{
  "index": "2019-B-3",
  "type": "ALG",
  "tag": [
    "ALG",
    "ANA",
    "NT"
  ],
  "difficulty": "",
  "question": "Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u \\in \\mathbb{R}^n$ be a unit column vector (that is,\n$u^T u = 1$). Let $P = I - 2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.",
  "solution": "\\noindent\n\\textbf{Solution 1.}\nWe first note that $P$ corresponds to the linear transformation on $\\mathbb{R}^n$ given by reflection in the hyperplane perpendicular to $u$: $P(u) = -u$, and for any $v$ with $\\langle u,v\\rangle = 0$, $P(v) = v$. In particular, $P$ is an orthogonal matrix of determinant $-1$.\n\nWe next claim that if $Q$ is an $n\\times n$ orthogonal matrix that does not have $1$ as an eigenvalue, then $\\det Q = (-1)^n$. To see this, recall that the roots of the characteristic polynomial $p(t) = \\det(tI-Q)$ all lie on the unit circle in $\\mathbb{C}$, and all non-real roots occur in conjugate pairs ($p(t)$ has real coefficients, and orthogonality implies that $p(t) = \\pm t^n p(t^{-1})$). The product of each conjugate pair of roots is $1$; thus $\\det Q = (-1)^k$ where $k$ is the multiplicity of $-1$ as a root of $p(t)$. Since $1$ is not a root and all other roots appear in conjugate pairs, $k$ and $n$ have the same parity, and so $\\det Q = (-1)^n$.\n\nFinally, if neither of the orthogonal matrices $Q$ nor $PQ$ has $1$ as an eigenvalue, then $\\det Q = \\det(PQ) = (-1)^n$, contradicting the fact that $\\det P = -1$. The result follows.\n\n\\noindent\n\\textbf{Remark.}\nIt can be shown that any $n \\times n$ orthogonal matrix $Q$ can be written as a product of at most $n$ hyperplane reflections (Householder matrices). If equality occurs, then $\\det(Q) = (-1)^n$;\nif equality does not occur, then $Q$ has $1$ as an eigenvalue.\nConsequently, equality fails for one of $Q$ and $PQ$, and that matrix has $1$ as an eigenvalue.\n\nSucharit Sarkar suggests the following topological interpretation: an orthogonal matrix without 1 as an eigenvalue\ninduces a fixed-point-free map from the $(n-1)$-sphere to itself, and the degree of such a map must be $(-1)^n$.\n\n\\noindent\n\\textbf{Solution 2.}\nThis solution uses the (reverse) \\emph{Cayley transform}: if $Q$ is an orthogonal matrix not having 1 as an eigenvalue, then\n\\[\nA = (I-Q)(I+Q)^{-1}\n\\]\nis a skew-symmetric matrix (that is, $A^T = -A$).\n\nSuppose then that $Q$ does not have $1$ as an eigenvalue.\nLet $V$ be the orthogonal complement of $u$ in $\\mathbb{R}^n$. On one hand, for $v \\in V$,\n\\[\n(I-Q)^{-1} (I - QP) v = (I-Q)^{-1} (I-Q)v = v.\n\\]\nOn the other hand,\n\\[\n(I-Q)^{-1} (I - QP) u = (I-Q)^{-1} (I+Q)u = Au\n\\]\nand $\\langle u, Au \\rangle = \\langle A^T u, u \\rangle\n= \\langle -Au, u \\rangle$, so $Au \\in V$.\nPut $w = (1-A)u$; then $(1-QP)w = 0$, so $QP$ has 1 as an eigenvalue, and the same for $PQ$ because $PQ$ and $QP$ have the same characteristic polynomial.\n\n\\noindent\n\\textbf{Remark.}\nThe \\emph{Cayley transform} is the following construction: if $A$ is a skew-symmetric matrix,\nthen $I+A$ is invertible and\n\\[\nQ = (I-A)(I+A)^{-1}\n\\]\nis an orthogonal matrix.\n\n\\noindent\n\\textbf{Remark.}\n(by Steven Klee)\nA related argument is to compute $\\det(PQ-I)$ using the \\emph{matrix determinant lemma}:\nif $A$ is an invertible $n \\times n$ matrix and $v, w$ are $1 \\times n$ column vectors, then\n\\[\n\\det(A + vw^T) = \\det(A) (1 + w^T A^{-1} v).\n\\]\nThis reduces to the case $A = I$, in which case it again comes down to the fact that the product of two square matrices (in this case, obtained from $v$ and $w$ by padding with zeroes) retains the same characteristic polynomial when the factors are reversed.",
  "vars": [
    "t",
    "v",
    "w",
    "k",
    "p",
    "A"
  ],
  "params": [
    "n",
    "Q",
    "u",
    "P",
    "I",
    "V",
    "R",
    "C"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "t": "timevar",
        "v": "vectorv",
        "w": "vectorw",
        "k": "counterk",
        "p": "polyvar",
        "A": "skewmtx",
        "n": "dimens",
        "Q": "orthomat",
        "u": "unitvec",
        "P": "reflect",
        "I": "identity",
        "V": "orthosub",
        "R": "realnums",
        "C": "complex"
      },
      "question": "Let $orthomat$ be an $dimens$-by-$dimens$ real orthogonal matrix, and let $unitvec \\in \\mathbb{realnums}^{dimens}$ be a unit column vector (that is,\n$unitvec^T unitvec = 1$). Let $reflect = identity - 2 unitvec unitvec^T$, where $identity$ is the $dimens$-by-$dimens$ identity matrix. Show that if $1$ is not an eigenvalue of $orthomat$, then $1$ is an eigenvalue of $reflect\\,orthomat$.",
      "solution": "\\noindent\n\\textbf{Solution 1.}\nWe first note that $reflect$ corresponds to the linear transformation on $\\mathbb{realnums}^{dimens}$ given by reflection in the hyperplane perpendicular to $unitvec$: $reflect(unitvec)=-unitvec$, and for any $vectorv$ with $\\langle unitvec,vectorv\\rangle =0$, $reflect(vectorv)=vectorv$. In particular, $reflect$ is an orthogonal matrix of determinant $-1$.\n\nWe next claim that if $orthomat$ is a $dimens\\times dimens$ orthogonal matrix that does not have $1$ as an eigenvalue, then $\\det orthomat = (-1)^{dimens}$. To see this, recall that the roots of the characteristic polynomial $polyvar(timevar)=\\det(timevar\\,identity-orthomat)$ all lie on the unit circle in $\\mathbb{complex}$, and all non-real roots occur in conjugate pairs ($polyvar(timevar)$ has real coefficients, and orthogonality implies that $polyvar(timevar)=\\pm timevar^{dimens} polyvar(timevar^{-1})$). The product of each conjugate pair of roots is $1$; thus $\\det orthomat = (-1)^{counterk}$ where $counterk$ is the multiplicity of $-1$ as a root of $polyvar(timevar)$. Since $1$ is not a root and all other roots appear in conjugate pairs, $counterk$ and $dimens$ have the same parity, and so $\\det orthomat = (-1)^{dimens}$.\n\nFinally, if neither of the orthogonal matrices $orthomat$ nor $reflect\\,orthomat$ has $1$ as an eigenvalue, then $\\det orthomat = \\det(reflect\\,orthomat)=(-1)^{dimens}$, contradicting the fact that $\\det reflect = -1$. The result follows.\n\n\\noindent\n\\textbf{Remark.}\nIt can be shown that any $dimens \\times dimens$ orthogonal matrix $orthomat$ can be written as a product of at most $dimens$ hyperplane reflections (Householder matrices). If equality occurs, then $\\det(orthomat)=(-1)^{dimens}$; if equality does not occur, then $orthomat$ has $1$ as an eigenvalue. Consequently, equality fails for one of $orthomat$ and $reflect\\,orthomat$, and that matrix has $1$ as an eigenvalue.\n\nSucharit Sarkar suggests the following topological interpretation: an orthogonal matrix without $1$ as an eigenvalue induces a fixed-point-free map from the $(dimens-1)$-sphere to itself, and the degree of such a map must be $(-1)^{dimens}$.\n\n\\noindent\n\\textbf{Solution 2.}\nThis solution uses the (reverse) \\emph{Cayley transform}: if $orthomat$ is an orthogonal matrix not having $1$ as an eigenvalue, then\n\\[\nskewmtx =(identity-orthomat)(identity+orthomat)^{-1}\n\\]\nis a skew-symmetric matrix (that is, $skewmtx^T=-skewmtx$).\n\nSuppose then that $orthomat$ does not have $1$ as an eigenvalue.\nLet $orthosub$ be the orthogonal complement of $unitvec$ in $\\mathbb{realnums}^{dimens}$. On one hand, for $vectorv \\in orthosub$,\n\\[\n(identity-orthomat)^{-1}(identity-orthomat\\,reflect)\\,vectorv=(identity-orthomat)^{-1}(identity-orthomat)vectorv=vectorv.\n\\]\nOn the other hand,\n\\[\n(identity-orthomat)^{-1}(identity-orthomat\\,reflect)\\,unitvec=(identity-orthomat)^{-1}(identity+orthomat)unitvec=skewmtx\\,unitvec,\n\\]\nand $\\langle unitvec,skewmtx\\,unitvec\\rangle=\\langle skewmtx^T unitvec,unitvec\\rangle=\\langle -skewmtx\\,unitvec,unitvec\\rangle$, so $skewmtx\\,unitvec\\in orthosub$. Put $vectorw=(1-skewmtx)unitvec$; then $(1-orthomat\\,reflect)vectorw=0$, so $orthomat\\,reflect$ has $1$ as an eigenvalue, and the same for $reflect\\,orthomat$ because $reflect\\,orthomat$ and $orthomat\\,reflect$ have the same characteristic polynomial.\n\n\\noindent\n\\textbf{Remark.}\nThe \\emph{Cayley transform} is the following construction: if $skewmtx$ is a skew-symmetric matrix, then $identity+skewmtx$ is invertible and\n\\[\northomat=(identity-skewmtx)(identity+skewmtx)^{-1}\n\\]\nis an orthogonal matrix.\n\n\\noindent\n\\textbf{Remark.}\n(by Steven Klee)\nA related argument is to compute $\\det(reflect\\,orthomat-identity)$ using the \\emph{matrix determinant lemma}: if $skewmtx$ is an invertible $dimens \\times dimens$ matrix and $vectorv,vectorw$ are $1 \\times dimens$ column vectors, then\n\\[\n\\det(skewmtx+vectorv vectorw^T)=\\det(skewmtx) \\bigl(1+vectorw^T skewmtx^{-1} vectorv\\bigr).\n\\]\nThis reduces to the case $skewmtx=identity$, in which case it again comes down to the fact that the product of two square matrices (in this case, obtained from $vectorv$ and $vectorw$ by padding with zeros) retains the same characteristic polynomial when the factors are reversed."
    },
    "descriptive_long_confusing": {
      "map": {
        "t": "pebblestone",
        "v": "lighthouse",
        "w": "buttercup",
        "k": "horsewhip",
        "p": "crocodile",
        "A": "snowflurry",
        "n": "jellybean",
        "Q": "dragonfly",
        "u": "raincloud",
        "P": "thunderbolt",
        "I": "moonlight",
        "V": "sandcastle",
        "R": "blueberries",
        "C": "marshmallow"
      },
      "question": "Let $dragonfly$ be an $jellybean$-by-$jellybean$ real orthogonal matrix, and let $raincloud \\in \\mathbb{blueberries}^{jellybean}$ be a unit column vector (that is,\n$raincloud^T raincloud = 1$). Let $thunderbolt = moonlight - 2raincloud raincloud^T$, where $moonlight$ is the $jellybean$-by-$jellybean$ identity matrix. Show that if $1$ is not an eigenvalue of $dragonfly$, then $1$ is an eigenvalue of $thunderbolt dragonfly$.",
      "solution": "\\noindent\n\\textbf{Solution 1.}\nWe first note that $thunderbolt$ corresponds to the linear transformation on $\\mathbb{blueberries}^{jellybean}$ given by reflection in the hyperplane perpendicular to $raincloud$: $thunderbolt(raincloud) = -raincloud$, and for any $lighthouse$ with $\\langle raincloud,lighthouse\\rangle = 0$, $thunderbolt(lighthouse) = lighthouse$. In particular, $thunderbolt$ is an orthogonal matrix of determinant $-1$.\n\nWe next claim that if $dragonfly$ is an $jellybean\\times jellybean$ orthogonal matrix that does not have $1$ as an eigenvalue, then $\\det dragonfly = (-1)^{jellybean}$. To see this, recall that the roots of the characteristic polynomial $crocodile(pebblestone) = \\det(pebblestone moonlight-dragonfly)$ all lie on the unit circle in $\\mathbb{marshmallow}$, and all non-real roots occur in conjugate pairs ($crocodile(pebblestone)$ has real coefficients, and orthogonality implies that $crocodile(pebblestone) = \\pm pebblestone^{jellybean} crocodile(pebblestone^{-1})$). The product of each conjugate pair of roots is $1$; thus $\\det dragonfly = (-1)^{horsewhip}$ where $horsewhip$ is the multiplicity of $-1$ as a root of $crocodile(pebblestone)$. Since $1$ is not a root and all other roots appear in conjugate pairs, $horsewhip$ and $jellybean$ have the same parity, and so $\\det dragonfly = (-1)^{jellybean}$.\n\nFinally, if neither of the orthogonal matrices $dragonfly$ nor $thunderbolt dragonfly$ has $1$ as an eigenvalue, then $\\det dragonfly = \\det(thunderbolt dragonfly) = (-1)^{jellybean}$, contradicting the fact that $\\det thunderbolt = -1$. The result follows.\n\n\\noindent\n\\textbf{Remark.}\nIt can be shown that any $jellybean \\times jellybean$ orthogonal matrix $dragonfly$ can be written as a product of at most $jellybean$ hyperplane reflections (Householder matrices). If equality occurs, then $\\det(dragonfly) = (-1)^{jellybean}$;\nif equality does not occur, then $dragonfly$ has $1$ as an eigenvalue.\nConsequently, equality fails for one of $dragonfly$ and $thunderbolt dragonfly$, and that matrix has $1$ as an eigenvalue.\n\nSucharit Sarkar suggests the following topological interpretation: an orthogonal matrix without 1 as an eigenvalue\ninduces a fixed-point-free map from the $(jellybean-1)$-sphere to itself, and the degree of such a map must be $(-1)^{jellybean}$.\n\n\\noindent\n\\textbf{Solution 2.}\nThis solution uses the (reverse) \\emph{Cayley transform}: if $dragonfly$ is an orthogonal matrix not having 1 as an eigenvalue, then\n\\[\nsnowflurry = (moonlight-dragonfly)(moonlight+dragonfly)^{-1}\n\\]\nis a skew-symmetric matrix (that is, $snowflurry^T = -snowflurry$).\n\nSuppose then that $dragonfly$ does not have $1$ as an eigenvalue.\nLet $sandcastle$ be the orthogonal complement of $raincloud$ in $\\mathbb{blueberries}^{jellybean}$. On one hand, for $lighthouse \\in sandcastle$,\n\\[\n(moonlight-dragonfly)^{-1} (moonlight - dragonfly thunderbolt) lighthouse = (moonlight-dragonfly)^{-1} (moonlight-dragonfly)lighthouse = lighthouse.\n\\]\nOn the other hand,\n\\[\n(moonlight-dragonfly)^{-1} (moonlight - dragonfly thunderbolt) raincloud = (moonlight-dragonfly)^{-1} (moonlight+dragonfly)raincloud = snowflurry raincloud\n\\]\nand $\\langle raincloud, snowflurry raincloud \\rangle = \\langle snowflurry^T raincloud, raincloud \\rangle\n= \\langle -snowflurry raincloud, raincloud \\rangle$, so $snowflurry raincloud \\in sandcastle$.\nPut $buttercup = (1-snowflurry)raincloud$; then $(1-dragonfly thunderbolt)buttercup = 0$, so $dragonfly thunderbolt$ has 1 as an eigenvalue, and the same for $thunderbolt dragonfly$ because $thunderbolt dragonfly$ and $dragonfly thunderbolt$ have the same characteristic polynomial.\n\n\\noindent\n\\textbf{Remark.}\nThe \\emph{Cayley transform} is the following construction: if $snowflurry$ is a skew-symmetric matrix,\nthen $moonlight+snowflurry$ is invertible and\n\\[\ndragonfly = (moonlight-snowflurry)(moonlight+snowflurry)^{-1}\n\\]\nis an orthogonal matrix.\n\n\\noindent\n\\textbf{Remark.}\n(by Steven Klee)\nA related argument is to compute $\\det(thunderbolt dragonfly-moonlight)$ using the \\emph{matrix determinant lemma}:\nif $snowflurry$ is an invertible $jellybean \\times jellybean$ matrix and $lighthouse, buttercup$ are $1 \\times jellybean$ column vectors, then\n\\[\n\\det(snowflurry + lighthouse buttercup^T) = \\det(snowflurry) (1 + buttercup^T snowflurry^{-1} lighthouse).\n\\]\nThis reduces to the case $snowflurry = moonlight$, in which case it again comes down to the fact that the product of two square matrices (in this case, obtained from $lighthouse$ and $buttercup$ by padding with zeroes) retains the same characteristic polynomial when the factors are reversed."
    },
    "descriptive_long_misleading": {
      "map": {
        "t": "timeless",
        "v": "scalarval",
        "w": "motionless",
        "k": "scarcity",
        "p": "constant",
        "A": "symmetry",
        "n": "dimensionless",
        "Q": "nonorthogonal",
        "u": "nullvector",
        "P": "rotation",
        "I": "zeromatrix",
        "V": "parallelspace",
        "R": "imaginaryset",
        "C": "realfield"
      },
      "question": "Let $nonorthogonal$ be an $dimensionless$-by-$dimensionless$ real orthogonal matrix, and let $nullvector \\in \\mathbb{imaginaryset}^{dimensionless}$ be a unit column vector (that is,\n$nullvector^T nullvector = 1$). Let $rotation = zeromatrix - 2nullvectornullvector^T$, where $zeromatrix$ is the $dimensionless$-by-$dimensionless$ identity matrix. Show that if $1$ is not an eigenvalue of $nonorthogonal$, then $1$ is an eigenvalue of $rotationnonorthogonal$.",
      "solution": "\\noindent\n\\textbf{Solution 1.}\nWe first note that $rotation$ corresponds to the linear transformation on $\\mathbb{imaginaryset}^{dimensionless}$ given by reflection in the hyperplane perpendicular to $nullvector$: $rotation(nullvector) = -nullvector$, and for any $scalarval$ with $\\langle nullvector,scalarval\\rangle = 0$, $rotation(scalarval) = scalarval$. In particular, $rotation$ is an orthogonal matrix of determinant $-1$.\n\nWe next claim that if $nonorthogonal$ is an $dimensionless\\times dimensionless$ orthogonal matrix that does not have $1$ as an eigenvalue, then $\\det nonorthogonal = (-1)^{dimensionless}$. To see this, recall that the roots of the characteristic polynomial $constant(t) = \\det(tzeromatrix-nonorthogonal)$ all lie on the unit circle in $\\mathbb{realfield}$, and all non-real roots occur in conjugate pairs ($constant(t)$ has real coefficients, and orthogonality implies that $constant(t) = \\pm t^{dimensionless} constant(t^{-1})$). The product of each conjugate pair of roots is $1$; thus $\\det nonorthogonal = (-1)^{scarcity}$ where $scarcity$ is the multiplicity of $-1$ as a root of $constant(t)$. Since $1$ is not a root and all other roots appear in conjugate pairs, $scarcity$ and $dimensionless$ have the same parity, and so $\\det nonorthogonal = (-1)^{dimensionless}$.\n\nFinally, if neither of the orthogonal matrices $nonorthogonal$ nor $rotationnonorthogonal$ has $1$ as an eigenvalue, then $\\det nonorthogonal = \\det(rotationnonorthogonal) = (-1)^{dimensionless}$, contradicting the fact that $\\det rotation = -1$. The result follows.\n\n\\noindent\n\\textbf{Remark.}\nIt can be shown that any $dimensionless \\times dimensionless$ orthogonal matrix $nonorthogonal$ can be written as a product of at most $dimensionless$ hyperplane reflections (Householder matrices). If equality occurs, then $\\det(nonorthogonal) = (-1)^{dimensionless}$;\nif equality does not occur, then $nonorthogonal$ has $1$ as an eigenvalue.\nConsequently, equality fails for one of $nonorthogonal$ and $rotationnonorthogonal$, and that matrix has $1$ as an eigenvalue.\n\nSucharit Sarkar suggests the following topological interpretation: an orthogonal matrix without 1 as an eigenvalue induces a fixed-point-free map from the $(dimensionless-1)$-sphere to itself, and the degree of such a map must be $(-1)^{dimensionless}$.\n\n\\noindent\n\\textbf{Solution 2.}\nThis solution uses the (reverse) \\emph{Cayley transform}: if $nonorthogonal$ is an orthogonal matrix not having 1 as an eigenvalue, then\n\\[\nsymmetry = (zeromatrix-nonorthogonal)(zeromatrix+nonorthogonal)^{-1}\n\\]\nis a skew-symmetric matrix (that is, $symmetry^T = -symmetry$).\n\nSuppose then that $nonorthogonal$ does not have $1$ as an eigenvalue.\nLet $parallelspace$ be the orthogonal complement of $nullvector$ in $\\mathbb{imaginaryset}^{dimensionless}$. On one hand, for $scalarval \\in parallelspace$,\n\\[\n(zeromatrix-nonorthogonal)^{-1} (zeromatrix - nonorthogonal rotation) scalarval = (zeromatrix-nonorthogonal)^{-1} (zeromatrix-nonorthogonal)scalarval = scalarval.\n\\]\nOn the other hand,\n\\[\n(zeromatrix-nonorthogonal)^{-1} (zeromatrix - nonorthogonal rotation) nullvector = (zeromatrix-nonorthogonal)^{-1} (zeromatrix+nonorthogonal)nullvector = symmetrynullvector\n\\]\nand $\\langle nullvector, symmetrynullvector \\rangle = \\langle symmetry^T nullvector, nullvector \\rangle = \\langle -symmetrynullvector, nullvector \\rangle$, so $symmetrynullvector \\in parallelspace$.\nPut $motionless = (1-symmetry)nullvector$; then $(1-nonorthogonal rotation)motionless = 0$, so $nonorthogonal rotation$ has 1 as an eigenvalue, and the same for $rotationnonorthogonal$ because $rotationnonorthogonal$ and $nonorthogonal rotation$ have the same characteristic polynomial.\n\n\\noindent\n\\textbf{Remark.}\nThe \\emph{Cayley transform} is the following construction: if $symmetry$ is a skew-symmetric matrix, then $zeromatrix+symmetry$ is invertible and\n\\[\nnonorthogonal = (zeromatrix-symmetry)(zeromatrix+symmetry)^{-1}\n\\]\nis an orthogonal matrix.\n\n\\noindent\n\\textbf{Remark.}\n(by Steven Klee)\nA related argument is to compute $\\det(rotationnonorthogonal-zeromatrix)$ using the \\emph{matrix determinant lemma}: if $symmetry$ is an invertible $dimensionless \\times dimensionless$ matrix and $scalarval, motionless$ are $1 \\times dimensionless$ column vectors, then\n\\[\n\\det(symmetry + scalarval motionless^T) = \\det(symmetry) (1 + motionless^T symmetry^{-1} scalarval).\n\\]\nThis reduces to the case $symmetry = zeromatrix$, in which case it again comes down to the fact that the product of two square matrices (in this case, obtained from $scalarval$ and $motionless$ by padding with zeroes) retains the same characteristic polynomial when the factors are reversed."
    },
    "garbled_string": {
      "map": {
        "t": "qzxwvtnp",
        "v": "hjgrksla",
        "w": "mdufpxye",
        "k": "brnlqvsc",
        "p": "zltehskq",
        "A": "csiodvma",
        "n": "xyqambdo",
        "Q": "dfkrujap",
        "u": "nbazmxle",
        "P": "ygehclir",
        "I": "ksdaptro",
        "V": "ruvpzaqe",
        "R": "xvyldtec",
        "C": "wpfgrnob"
      },
      "question": "Let $dfkrujap$ be an $xyqambdo$-by-$xyqambdo$ real orthogonal matrix, and let $nbazmxle \\in \\mathbb{R}^{xyqambdo}$ be a unit column vector (that is,\n$nbazmxle^T nbazmxle = 1$). Let $ygehclir = ksdaptro - 2nbazmxlenbazmxle^T$, where $ksdaptro$ is the $xyqambdo$-by-$xyqambdo$ identity matrix. Show that if $1$ is not an eigenvalue of $dfkrujap$, then $1$ is an eigenvalue of $ygehclirdfkrujap$.",
      "solution": "\\noindent\n\\textbf{Solution 1.}\nWe first note that $ygehclir$ corresponds to the linear transformation on $\\mathbb{R}^{xyqambdo}$ given by reflection in the hyperplane perpendicular to $nbazmxle$: $ygehclir(nbazmxle) = -nbazmxle$, and for any $hjgrksla$ with $\\langle nbazmxle,hjgrksla\\rangle = 0$, $ygehclir(hjgrksla) = hjgrksla$. In particular, $ygehclir$ is an orthogonal matrix of determinant $-1$.\n\nWe next claim that if $dfkrujap$ is an $xyqambdo\\times xyqambdo$ orthogonal matrix that does not have $1$ as an eigenvalue, then $\\det dfkrujap = (-1)^{xyqambdo}$. To see this, recall that the roots of the characteristic polynomial $zltehskq(qzxwvtnp) = \\det(qzxwvtnp ksdaptro-dfkrujap)$ all lie on the unit circle in $\\mathbb{C}$, and all non-real roots occur in conjugate pairs ($zltehskq(qzxwvtnp)$ has real coefficients, and orthogonality implies that $zltehskq(qzxwvtnp) = \\pm qzxwvtnp^{xyqambdo} zltehskq(qzxwvtnp^{-1})$). The product of each conjugate pair of roots is $1$; thus $\\det dfkrujap = (-1)^{brnlqvsc}$ where $brnlqvsc$ is the multiplicity of $-1$ as a root of $zltehskq(qzxwvtnp)$. Since $1$ is not a root and all other roots appear in conjugate pairs, $brnlqvsc$ and $xyqambdo$ have the same parity, and so $\\det dfkrujap = (-1)^{xyqambdo}$.\n\nFinally, if neither of the orthogonal matrices $dfkrujap$ nor $ygehclirdfkrujap$ has $1$ as an eigenvalue, then $\\det dfkrujap = \\det(ygehclirdfkrujap) = (-1)^{xyqambdo}$, contradicting the fact that $\\det ygehclir = -1$. The result follows.\n\n\\noindent\n\\textbf{Remark.}\nIt can be shown that any $xyqambdo \\times xyqambdo$ orthogonal matrix $dfkrujap$ can be written as a product of at most $xyqambdo$ hyperplane reflections (Householder matrices). If equality occurs, then $\\det(dfkrujap) = (-1)^{xyqambdo}$;\nif equality does not occur, then $dfkrujap$ has $1$ as an eigenvalue.\nConsequently, equality fails for one of $dfkrujap$ and $ygehclirdfkrujap$, and that matrix has $1$ as an eigenvalue.\n\nSucharit Sarkar suggests the following topological interpretation: an orthogonal matrix without 1 as an eigenvalue\ninduces a fixed-point-free map from the $(xyqambdo-1)$-sphere to itself, and the degree of such a map must be $(-1)^{xyqambdo}$.\n\n\\noindent\n\\textbf{Solution 2.}\nThis solution uses the (reverse) \\emph{Cayley transform}: if $dfkrujap$ is an orthogonal matrix not having 1 as an eigenvalue, then\n\\[\ncsiodvma = (ksdaptro-dfkrujap)(ksdaptro+dfkrujap)^{-1}\n\\]\nis a skew-symmetric matrix (that is, $csiodvma^T = -csiodvma$).\n\nSuppose then that $dfkrujap$ does not have $1$ as an eigenvalue.\nLet $ruvpzaqe$ be the orthogonal complement of $nbazmxle$ in $\\mathbb{R}^{xyqambdo}$. On one hand, for $hjgrksla \\in ruvpzaqe$,\n\\[\n(ksdaptro-dfkrujap)^{-1} (ksdaptro-dfkrujap ygehclir) hjgrksla = (ksdaptro-dfkrujap)^{-1} (ksdaptro-dfkrujap) hjgrksla = hjgrksla.\n\\]\nOn the other hand,\n\\[\n(ksdaptro-dfkrujap)^{-1} (ksdaptro-dfkrujap ygehclir) nbazmxle = (ksdaptro-dfkrujap)^{-1} (ksdaptro+dfkrujap) nbazmxle = csiodvma nbazmxle\n\\]\nand $\\langle nbazmxle, csiodvma nbazmxle \\rangle = \\langle csiodvma^T nbazmxle, nbazmxle \\rangle\n= \\langle -csiodvma nbazmxle, nbazmxle \\rangle$, so $csiodvma nbazmxle \\in ruvpzaqe$.\nPut $mdufpxye = (1-csiodvma) nbazmxle$; then $(1-dfkrujapygehclir) mdufpxye = 0$, so $dfkrujapygehclir$ has 1 as an eigenvalue, and the same for $ygehclirdfkrujap$ because $ygehclirdfkrujap$ and $dfkrujapygehclir$ have the same characteristic polynomial.\n\n\\noindent\n\\textbf{Remark.}\nThe \\emph{Cayley transform} is the following construction: if $csiodvma$ is a skew-symmetric matrix,\nthen $ksdaptro+csiodvma$ is invertible and\n\\[\ndfkrujap = (ksdaptro-csiodvma)(ksdaptro+csiodvma)^{-1}\\]\nis an orthogonal matrix.\n\n\\noindent\n\\textbf{Remark.}\n(by Steven Klee)\nA related argument is to compute $\\det(ygehclirdfkrujap-ksdaptro)$ using the \\emph{matrix determinant lemma}:\nif $csiodvma$ is an invertible $xyqambdo \\times xyqambdo$ matrix and $hjgrksla, mdufpxye$ are $1 \\times xyqambdo$ column vectors, then\n\\[\n\\det(csiodvma + hjgrkslamdufpxye^T) = \\det(csiodvma) (1 + mdufpxye^T csiodvma^{-1} hjgrksla).\n\\]\nThis reduces to the case $csiodvma = ksdaptro$, in which case it again comes down to the fact that the product of two square matrices (in this case, obtained from $hjgrksla$ and $mdufpxye$ by padding with zeroes) retains the same characteristic polynomial when the factors are reversed."
    },
    "kernel_variant": {
      "question": "Let $n\\ge 3$ and let $Q\\in O(n,\\mathbb R)$ be an orthogonal $n\\times n$ matrix such that  \n$1\\not\\in\\sigma(Q)$.  \nFix an \\emph{odd} integer $d$ with $1\\le d\\le n-1$ and choose a full–column–rank matrix $U\\in\\mathbb R^{\\,n\\times d}$.  \nPut  \n\\[\n        P:=I_n-2\\,U\\bigl(U^{\\!T}U\\bigr)^{-1}U^{\\!T},\n        \\qquad \n        W:=\\operatorname{Im}U,\n\\tag{$*$}\n\\]\nso that $P$ is the Householder reflection in the $d$–dimensional subspace $W$.  Set  \n\\[\n        T:=Q\\,P .\n\\]\n\n\\begin{enumerate}\n\\item[(a)]  Prove that $P\\in O(n,\\mathbb R)$, that $\\det P=(-1)^{d}$, and that $P\\!\\!\\mid_{W}=-I_{W}$ while\n            $P\\!\\!\\mid_{W^{\\perp}}=I_{W^{\\perp}}$.\n\n\\item[(b)]  Show that $1$ is an eigenvalue of $T$.\n\n\\item[(c)]  Write $\\mathbb R^{n}=W\\oplus W^{\\perp}$ and\n            \\[\n                   Q=\n                   \\begin{bmatrix}\n                       A&B\\\\\n                       C&D\n                   \\end{bmatrix},\n                   \\qquad\n                   A\\in\\mathbb R^{d\\times d},\\;\n                   B\\in\\mathbb R^{d\\times(n-d)},\\;\n                   C\\in\\mathbb R^{(n-d)\\times d},\\;\n                   D\\in\\mathbb R^{(n-d)\\times(n-d)} .\n            \\]\n            \\begin{enumerate}\n            \\item[(i)]  Show that $I_{\\,n-d}-D$ is invertible and define\n                         \\[\n                               R:=(I_{\\,n-d}-D)^{-1},\\qquad  \n                               S:=I_d+A+BR\\,C .\n                         \\]\n                         Put further\n                         \\[\n                               E:=I_d-A-BR\\,C ,\\qquad\n                               J:=E^{-1}S .\n                         \\]\n                         Prove explicitly that $J^{\\!T}=-\\,J$ and deduce from this that $S$ is singular whenever $d$ is odd.\n\n            \\item[(ii)]  For every $x\\in\\ker S$ define\n                         \\[\n                               w(x):=\n                               \\begin{bmatrix}\n                                     x\\\\[4pt]\n                                     -\\,R\\,C\\,x\n                               \\end{bmatrix}.\n                         \\]\n                         Show that $T\\,w(x)=w(x)$.  Conclude that\n                         $\\dim\\ker(I_n-T)\\ge 1$ and that the algebraic\n                         multiplicity of the eigenvalue $1$ in $T$\n                         has the same parity as $d$.\n            \\end{enumerate}\n\n\\item[(d)]  Describe an explicit algorithm that writes an arbitrary\n            $Q\\in O(n,\\mathbb R)$ as a product of \\emph{at most $n$}\n            genuine (rank–one and different from $I_n$) Householder reflections\n            $I_n-2\\,vv^{\\!T}/(v^{\\!T}v)$, and prove that $-I_n$\n            requires \\emph{exactly} $n$ such reflections.\n            Hence the Cartan–Dieudonné bound $n$ is sharp.\n\n\\item[(e)]  Let $r(n)$ be the minimal integer such that every element\n            of $SO(n)$ is a product of $r(n)$ rank–one Householder\n            reflections.  Prove that\n            \\[\n                 r(n)=\n                 \\begin{cases}\n                       n,   & n\\ \\text{even},\\\\[6pt]\n                       n-1, & n\\ \\text{odd}.\n                 \\end{cases}\n            \\]\n\\end{enumerate}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Throughout $I_k$ denotes the $k\\times k$ identity matrix; all matrices are real.\n\n\\vspace{6pt}\n\\textbf{(a)  Elementary properties of $P$.}  \nBecause $U$ has full column rank, $G:=U^{\\!T}U$ is symmetric positive–definite and therefore invertible.  Direct computation gives  \n\\[\n     P^{\\!T}P\n     =(I_n-2UG^{-1}U^{\\!T})^{\\!T}(I_n-2UG^{-1}U^{\\!T})\n     =I_n ,\n\\]\nso $P\\in O(n,\\mathbb R)$.  \n\nChoose an orthonormal basis\n$\\{e_1,\\dots,e_d\\}$ of $W$ and extend it to\n$\\{e_1,\\dots,e_d,f_1,\\dots,f_{\\,n-d}\\}$ of $\\mathbb R^{n}$.\nRelative to this basis\n\\[\n      P=\\operatorname{diag}\\!\\bigl(-I_d,\\;I_{\\,n-d}\\bigr),\n\\]\nwhence $\\det P=(-1)^d$ and\n$P\\!\\!\\mid_{W}=-I_{W}$, $P\\!\\!\\mid_{W^{\\perp}}=I_{W^{\\perp}}$.\n\n\\vspace{6pt}\n\\textbf{(b)  A determinant argument.}  \n\nBecause $d$ is odd, part (a) gives $\\det P=-1$.  Hence\n\\[\n      \\det T\n      =\\det(QP)\n      =(\\det Q)(\\det P)\n      =-\\det Q .\n\\tag{1}\n\\]\nAssume for a contradiction that $1\\not\\in\\sigma(T)$.\nFor a real orthogonal matrix whose spectrum avoids $1$, every eigenvalue is either $-1$ or comes in a complex conjugate pair\n$\\{\\lambda,\\overline{\\lambda}\\}$ with $\\lambda\\overline{\\lambda}=1$; therefore  \n$\\det T=\\det Q=(-1)^{n}$.  This contradicts (1), so $1\\in\\sigma(T)$.\n\n\\vspace{6pt}\n\\textbf{(c)  Block computations.}\n\nWe decompose $Q$ with respect to the splitting\n$\\mathbb R^{n}=W\\oplus W^{\\perp}$:\n\\[\n   Q=\n   \\begin{bmatrix}\n         A&B\\\\ C&D\n   \\end{bmatrix}.\n\\]\nOrthogonality of $Q$ entails\n\\begin{equation}\n\\begin{aligned}\n   A^{\\!T}A+C^{\\!T}C &= I_d,\\\\\n   B^{\\!T}B+D^{\\!T}D &= I_{\\,n-d},\\\\\n   A^{\\!T}B+C^{\\!T}D &= 0 .\n\\end{aligned}\\tag{2}\n\\end{equation}\n\n\\smallskip\n\\emph{(i)  The matrices $R,\\,S,\\,E$ and a skew–symmetric block.}  \n\n\\underline{Invertibility of $I_{\\,n-d}-D$.}  \nIf $(I_{\\,n-d}-D)y=0$ with $y\\neq 0$, then $Dy=y$.  From the second line of (2),\n\\[\n      \\lVert By\\rVert^{2}=y^{\\!T}B^{\\!T}B y\n                        =y^{\\!T}\\bigl(I_{\\,n-d}-D^{\\!T}D\\bigr)y\n                        =0 ,\n\\]\nso $By=0$.  Consequently\n$Q\\!\\begin{bmatrix}0\\\\ y\\end{bmatrix}\n   =\\begin{bmatrix}0\\\\ y\\end{bmatrix}$,\ncontradicting $1\\not\\in\\sigma(Q)$.  Therefore $I_{\\,n-d}-D$ is invertible and $R$ is well–defined.\n\n\\medskip\n\\underline{The Cayley transform.}  \nBecause $1\\not\\in\\sigma(Q)$, the Cayley matrix\n\\[\n       K:=(I_n-Q)^{-1}(I_n+Q)\n\\tag{3}\n\\]\nis defined.  \nA direct calculation shows\n\\[\n   K^{\\!T}\n   =\\bigl((I_n+Q)^{\\!T}\\bigr)(I_n-Q)^{-T}\n   =(I_n+Q^{\\!T})(I_n-Q^{\\!T})^{-1}\n   =-(I_n-Q)^{-1}(I_n+Q)\n   =-K ,\n\\]\nso $K^{\\!T}=-K$.\n\n\\smallskip\nWrite\n\\[\n   I_n-Q=\n   \\begin{bmatrix}\n         I_d-A & -B\\\\\n         -C    & I_{\\,n-d}-D\n   \\end{bmatrix},\n   \\qquad\n   I_n+Q=\n   \\begin{bmatrix}\n         I_d+A &  B\\\\\n         C     & I_{\\,n-d}+D\n   \\end{bmatrix}.\n\\]\nBecause $I_{\\,n-d}-D=R^{-1}$ is invertible, the inverse of $I_n-Q$\ncan be expressed through its Schur complement\n\\[\n      E:=I_d-A-BR\\,C .\n\\tag{4}\n\\]\n\\emph{Block inversion.}  \nWith $A_0:=I_d-A$, $B_0:=-B$, $C_0:=-C$, $D_0:=R^{-1}$,  \nthe inverse of\n$\\begin{bmatrix}A_0&B_0\\\\ C_0&D_0\\end{bmatrix}$\nis\n\\[\n   \\begin{bmatrix}\n      E^{-1} & -E^{-1}B_0D_0^{-1}\\\\[4pt]\n      -D_0^{-1}C_0E^{-1} & D_0^{-1}+D_0^{-1}C_0E^{-1}B_0D_0^{-1}\n   \\end{bmatrix},\n\\]\nwhich here becomes\n\\[\n   (I_n-Q)^{-1}=\n   \\begin{bmatrix}\n      E^{-1} & \\;E^{-1}B R\\\\[4pt]\n      \\;R C E^{-1} & R+R C E^{-1}B R\n   \\end{bmatrix}.\n\\tag{5}\n\\]\n(The lower–right block is now \\emph{plus}, correcting the sign error pointed out in the review.)\n\nMultiplying (5) by $I_n+Q$ we obtain\n\\[\n      K=\n      \\begin{bmatrix}\n          E^{-1}(I_d+A+BR\\,C) & *\\\\\n          *                   & *\n      \\end{bmatrix}.\n\\tag{6}\n\\]\nThus\n\\[\n      K_{11}=E^{-1}S=:J .\n\\tag{7}\n\\]\n\nSince $K^{\\!T}=-K$, each diagonal block of $K$ is itself\nskew–symmetric; hence\n\\[\n      J^{\\!T}=-J .\n\\tag{8}\n\\]\n\n\\medskip\n\\underline{Singularity of $S$.}  \nBecause $I_n-Q$ is invertible, its Schur complement $E$ is invertible.  From (7) we have\n$\\det S=\\det(E)\\det J$.  \nNow $d$ is odd and $J$ is skew–symmetric, so\n$\\det J=0$; hence $\\det S=0$ and $S$ is singular.\n\n\\smallskip\n\\emph{(ii)  Construction of $+1$–eigenvectors of $T$ and parity of their algebraic multiplicity.}  \n\nRelative to $W\\oplus W^{\\perp}$ we have\n\\[\n      P=\\operatorname{diag}\\!\\bigl(-I_{d},\\,I_{\\,n-d}\\bigr),\\qquad\n      T=QP=\n      \\begin{bmatrix}\n          -A & B\\\\\n          -C & D\n      \\end{bmatrix}.\n\\tag{9}\n\\]\n\nPick any non–zero $x\\in\\ker S$ and set\n\\[\n      y:=-R\\,C\\,x .\n\\tag{10}\n\\]\nDefine $w(x)$ as in the statement, i.e.\\ $w(x)=\\bigl[x^{\\!T},\\,y^{\\!T}\\bigr]^{\\!T}$.  \nUsing $y=-R\\,C\\,x$ and the identity \n\\[\n      D R = R-I_{\\,n-d},\n\\tag{11}\n\\]\nwe compute\n\\[\n   Tw(x)=\n   \\begin{bmatrix}\n          -A & B\\\\ -C & D\n   \\end{bmatrix}\n   \\begin{bmatrix}\n          x\\\\ y\n   \\end{bmatrix}\n   =\n   \\begin{bmatrix}\n          -A x + B y\\\\[2pt]\n          -C x + D y\n   \\end{bmatrix}.\n\\]\nThe lower block equals\n\\[\n    -C x + D y\n    = -C x - D R C x\n    = -C x - (R-I_{\\,n-d}) C x\n    = -C x - R C x + C x\n    = y ,\n\\]\nwhile the upper block simplifies to\n\\[\n     -A x + B y\n     = -A x - B R C x\n     = -\\bigl(A + B R C + I_d\\bigr)x + x\n     = x ,\n\\]\nbecause $x\\in\\ker S$ precisely means $(I_d+A+BR\\,C)x=0$.  Hence\n$Tw(x)=w(x)$, and $w(x)\\neq 0$.\n\nInjectivity of $x\\mapsto w(x)$ implies \n\\[\n      \\dim\\ker(I_n-T)\\ge\\dim\\ker S\\ge 1 .\n\\]\n\n\\underline{Parity of the algebraic multiplicity.}  \nLet  \n\\[\n      r:=\\text{algebraic multiplicity of }1\\text{ in }T, \\qquad\n      s:=\\text{algebraic multiplicity of }-1\\text{ in }T .\n\\]\nAll other eigenvalues occur in complex conjugate pairs on the unit\ncircle, and each such pair contributes an even number to the degree of\nthe characteristic polynomial.  Consequently\n\\[\n      r+s\\equiv n \\pmod 2 .\n\\tag{12}\n\\]\n\nThe determinant of an orthogonal matrix is the product of its\neigenvalues, hence  \n\\[\n      \\det T = (-1)^{s}.\n\\tag{13}\n\\]\nOn the other hand\n\\[\n      \\det T = \\det(Q)\\det(P) = (-1)^{n}\\,(-1)^{d}=(-1)^{\\,n+d},\n\\tag{14}\n\\]\nbecause (as observed in part (b)) $1\\not\\in\\sigma(Q)$ implies\n$\\det Q=(-1)^{n}$.  Comparing (13) with (14) gives\n\\[\n      s\\equiv n+d\\pmod 2 .\n\\tag{15}\n\\]\nCombining (12) and (15) yields\n\\[\n      r\\equiv d\\pmod 2 .\n\\]\nThus the algebraic multiplicity of the eigenvalue $1$ in $T$ has the same parity as $d$, as required.\n\n\\vspace{6pt}\n\\textbf{(d)  A constructive Cartan–Dieudonné factorisation.}\n\n\\emph{Algorithm.}  \nLet $Q_0:=Q$ and for $k=1,\\dots,n$ do:\n\n\\smallskip\n\\emph{Step $k$.}  \nIf $Q_{k-1}e_k=e_k$, do nothing;  \notherwise put\n\\[\n      u_k:=Q_{k-1}e_k-e_k\\neq 0,\\qquad\n      P_k:=I_n-\\frac{2\\,u_k u_k^{\\!T}}{u_k^{\\!T}u_k},\\qquad\n      Q_k:=P_kQ_{k-1}.\n\\]\nBecause $e_j^{\\!T}u_k=0$ for every $j<k$, the first $k-1$ columns of\n$Q_{k-1}$ remain unchanged and $Q_ke_k=e_k$.  After step $k$ the first\n$k$ columns of $Q_k$ coincide with those of $I_n$; consequently\n$Q_n=I_n$ and\n\\[\n      Q=P_1P_2\\cdots P_s ,\n\\tag{16}\n\\]\nwhere $s\\le n$ is the number of non–trivial steps.\nEach $P_k$ is a genuine rank–one Householder reflection, so at most\n$n$ such reflections are required.\n\n\\smallskip\n\\emph{Sharpness for $-I_n$.}  \nAssume $-I_n=R_1\\cdots R_m$ with rank–one reflections $R_j$.\nSince $\\det(-I_n)=(-1)^n$ and $\\det R_j=-1$, parity forces\n$m\\equiv n\\pmod 2$.  \nIf $m<n$, the $(n-1)$–dimensional fixed hyperplanes of the $R_j$\nwould have a non–trivial intersection, giving a non–zero vector fixed\nby $-I_n$, a contradiction.  Thus $m\\ge n$, and the bound $n$ in (16)\nis best possible.\n\n\\vspace{6pt}\n\\textbf{(e)  Minimal numbers inside $SO(n)$.}\n\n\\emph{Upper bounds.}  \nFactorisation (16) works for every $Q\\in SO(n)$ and uses at most\n$n$ reflections.  Because $\\det P_k=-1$ for each reflection,\nthe total number of reflections produced by the algorithm is\n\\emph{even}.  Hence\n\\[\n      r(n)\\le\n      \\begin{cases}\n            n,   & n\\ \\text{even},\\\\[6pt]\n            n-1, & n\\ \\text{odd}.\n      \\end{cases}\n\\tag{17}\n\\]\n\n\\emph{Lower bounds, $n$ even.}  \nFor $n=2m$ put\n\\[\n     Q_{\\mathrm{even}}\n     :=\\operatorname{diag}\\bigl(R(\\pi),\\dots,R(\\pi)\\bigr)\\in SO(n),\n     \\qquad\n     R(\\pi)=\n     \\begin{bmatrix}\n        -1&0\\\\ 0&-1\n     \\end{bmatrix}.\n\\]\nThus $Q_{\\mathrm{even}}=-I_n$, which part (d) showed requires exactly\n$n$ reflections, so $r(2m)\\ge 2m$.\n\n\\emph{Lower bounds, $n$ odd.}  \nFor $n=2m+1$ take\n\\[\n     Q_{\\mathrm{odd}}\n     :=\\operatorname{diag}\\bigl(R(\\pi),\\dots,R(\\pi),1\\bigr)\\in SO(n).\n\\]\nHere $Q_{\\mathrm{odd}}$ acts as $-I$ on an $(n-1)$–dimensional\nsubspace.  A product of fewer than $n-1$ reflections fixes at least a\n$2$–dimensional subspace, so it cannot be $Q_{\\mathrm{odd}}$.\nTherefore $r(2m+1)\\ge 2m$.\n\nCombining the bounds with (17) gives\n\\[\n      r(n)=\n      \\begin{cases}\n            n,   & n\\ \\text{even},\\\\[6pt]\n            n-1, & n\\ \\text{odd}.\n      \\end{cases}\n      \\qquad\\qquad\\square\n\\]\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.861394",
        "was_fixed": false,
        "difficulty_analysis": "• Higher–dimensional reflection:  The problem replaces a single\n  hyperplane reflection (d = 1) by a reflection through an arbitrary\n  odd-dimensional subspace, forcing the solver to cope with matrices of the\n  form I − 2U(UᵀU)⁻¹Uᵀ and to control their action on two complementary\n  subspaces simultaneously.\n\n• Quantitative bounds:  Part (b) does not merely assert the existence of\n  an eigenvalue 1; it demands a lower bound on its multiplicity, requiring\n  rank arguments that intertwine Q and P in a non-trivial way.\n\n• Interaction with the −1–eigenspace:  Part (c) brings in κ₋, the\n  multiplicity of −1 for Q, and asks for a sharp estimate.  The solver must\n  combine spectral information on Q with structural properties of P.\n\n• Factorisation problem:  Part (d) pushes the classical “every orthogonal\n  matrix is a product of reflections’’ into an optimisation question, whose\n  solution needs an iterative, dimension-counting construction.\n\n• Orientation constraint and SO(n):  Part (e) introduces parity issues for\n  determinants and forces a delicate analysis to obtain the exact number of\n  Householder reflections needed inside SO(n).\n\nAltogether these additions demand deeper linear-algebraic insight,\nspectral analysis, and combinatorial reasoning far beyond the original\nkernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let $n\\ge 3$ and let $Q\\in O(n,\\mathbb R)$ be an orthogonal $n\\times n$ matrix such that  \n$1\\not\\in\\sigma(Q)$.  \nFix an \\emph{odd} integer $d$ with $1\\le d\\le n-1$ and choose a full–column–rank matrix $U\\in\\mathbb R^{\\,n\\times d}$.  \nPut  \n\\[\n      P:=I_n-2\\,U\\bigl(U^{\\!T}U\\bigr)^{-1}U^{\\!T},\\qquad \n      W:=\\operatorname{Im}U ,\n\\tag{$*$}\n\\]\nso that $P$ is the Householder reflection in the $d$–dimensional subspace $W$.  Set  \n\\[\n      T:=Q\\,P .\n\\]\n\n\\begin{enumerate}\n\\item[(a)]  Prove that $P\\in O(n,\\mathbb R)$, that $\\det P=(-1)^{d}$, and that $P\\!\\!\\mid_{W}=-I_{W}$ while\n            $P\\!\\!\\mid_{W^{\\perp}}=I_{W^{\\perp}}$.\n\n\\item[(b)]  Show that $1$ is an eigenvalue of $T$.\n\n\\item[(c)]  Write $\\mathbb R^{n}=W\\oplus W^{\\perp}$ and\n            \\[\n                   Q=\n                   \\begin{bmatrix}\n                       A&B\\\\\n                       C&D\n                   \\end{bmatrix},\n                   \\qquad\n                   A\\in\\mathbb R^{d\\times d},\\;\n                   B\\in\\mathbb R^{d\\times(n-d)},\\;\n                   C\\in\\mathbb R^{(n-d)\\times d},\\;\n                   D\\in\\mathbb R^{(n-d)\\times(n-d)} .\n            \\]\n            \\begin{enumerate}\n            \\item[(i)]  Show that $I_{\\,n-d}-D$ is invertible and define\n                         \\[\n                               R:=(I_{\\,n-d}-D)^{-1},\\qquad  \n                               S:=I_d+A+BR\\,C .\n                         \\]\n                         Put further\n                         \\[\n                               E:=I_d-A-BR\\,C ,\\qquad\n                               J:=E^{-1}S .\n                         \\]\n                         Prove explicitly that $J^{\\!T}=-\\,J$ and deduce from this that $S$ is singular whenever $d$ is odd.\n\n            \\item[(ii)]  For every $x\\in\\ker S$ define\n                         \\[\n                               w(x):=\n                               \\begin{bmatrix}\n                                     x\\\\[4pt]\n                                     -\\,R\\,C\\,x\n                               \\end{bmatrix}.\n                         \\]\n                         Show that $T\\,w(x)=w(x)$.  Conclude that\n                         $\\dim\\ker(I_n-T)\\ge 1$ and that the algebraic\n                         multiplicity of the eigenvalue $1$ in $T$\n                         has the same parity as $d$.\n            \\end{enumerate}\n\n\\item[(d)]  Describe an explicit algorithm that writes an arbitrary\n            $Q\\in O(n,\\mathbb R)$ as a product of \\emph{at most $n$}\n            genuine (rank–one and different from $I_n$) Householder reflections\n            $I_n-2\\,vv^{\\!T}/(v^{\\!T}v)$, and prove that $-I_n$\n            requires \\emph{exactly} $n$ such reflections.\n            Hence the Cartan–Dieudonné bound $n$ is sharp.\n\n\\item[(e)]  Let $r(n)$ be the minimal integer such that every element\n            of $SO(n)$ is a product of $r(n)$ rank–one Householder\n            reflections.  Prove that\n            \\[\n                 r(n)=\n                 \\begin{cases}\n                       n,   & n\\ \\text{even},\\\\[6pt]\n                       n-1, & n\\ \\text{odd}.\n                 \\end{cases}\n            \\]\n\\end{enumerate}",
      "solution": "Throughout $I_k$ denotes the $k\\times k$ identity matrix; all matrices are real.\n\n\\vspace{6pt}\n\\textbf{(a)  Elementary properties of $P$.}  \nBecause $U$ has full column rank, $G:=U^{\\!T}U$ is symmetric positive–definite and therefore invertible.  Direct computation gives  \n\\[\n     P^{\\!T}P\n     =(I_n-2UG^{-1}U^{\\!T})^{\\!T}(I_n-2UG^{-1}U^{\\!T})\n     =I_n ,\n\\]\nso $P\\in O(n,\\mathbb R)$.  \n\nChoose an orthonormal basis\n$\\{e_1,\\dots,e_d\\}$ of $W$ and extend it to\n$\\{e_1,\\dots,e_d,f_1,\\dots,f_{\\,n-d}\\}$ of $\\mathbb R^{n}$.\nRelative to this basis\n\\[\n      P=\\operatorname{diag}\\bigl(-I_d,\\;I_{\\,n-d}\\bigr),\n\\]\nwhence $\\det P=(-1)^d$ and\n$P\\!\\!\\mid_{W}=-I_{W},\\;P\\!\\!\\mid_{W^{\\perp}}=I_{W^{\\perp}}$.\n\n\\vspace{6pt}\n\\textbf{(b)  A determinant argument.}  \n\nBecause $d$ is odd, part (a) gives $\\det P=-1$.  Hence\n\\[\n      \\det T\n      =\\det(QP)\n      =(\\det Q)(\\det P)\n      =-\\det Q .\n\\tag{1}\n\\]\nAssume for a contradiction that $1\\not\\in\\sigma(T)$.\nFor a real orthogonal matrix whose spectrum avoids $1$, every eigenvalue is either $-1$ or comes in a complex conjugate pair $\\{\\lambda,\\overline{\\lambda}\\}$ with $\\lambda\\overline{\\lambda}=1$; therefore  \n$\\det T=\\det Q=(-1)^{n}$, contradicting (1).  Thus $1\\in\\sigma(T)$.\n\n\\vspace{6pt}\n\\textbf{(c)  Block computations.}\n\nWe decompose $Q$ with respect to the splitting\n$\\mathbb R^{n}=W\\oplus W^{\\perp}$:\n\\[\n   Q=\n   \\begin{bmatrix}\n         A&B\\\\ C&D\n   \\end{bmatrix}.\n\\]\nOrthogonality of $Q$ entails\n\\begin{equation}\n\\begin{aligned}\n   A^{\\!T}A+C^{\\!T}C &= I_d,\\\\\n   B^{\\!T}B+D^{\\!T}D &= I_{\\,n-d},\\\\\n   A^{\\!T}B+C^{\\!T}D &= 0 .\n\\end{aligned}\\tag{2}\n\\end{equation}\n\n\\smallskip\n\\emph{(i)  The matrices $R,\\,S,\\,E$ and a skew–symmetric block.}  \n\n\\underline{Invertibility of $I_{\\,n-d}-D$.}  \nIf $(I_{\\,n-d}-D)y=0$ with $y\\neq 0$, then $Dy=y$.  From the first line of (2),\n\\[\n      \\lVert By\\rVert^{2}=y^{\\!T}B^{\\!T}B y\n                        =y^{\\!T}\\bigl(I_{\\,n-d}-D^{\\!T}D\\bigr)y\n                        =0 ,\n\\]\nso $By=0$.  Consequently\n$Q\\!\\begin{bmatrix}0\\\\ y\\end{bmatrix}\n   =\\begin{bmatrix}0\\\\ y\\end{bmatrix}$,\ncontradicting $1\\not\\in\\sigma(Q)$.  Therefore $I_{\\,n-d}-D$ is invertible and $R$ is well–defined.\n\n\\medskip\n\\underline{The Cayley transform.}  \nBecause $1\\not\\in\\sigma(Q)$, the Cayley matrix\n\\[\n       K:=(I_n-Q)^{-1}(I_n+Q)\n\\tag{3}\n\\]\nis defined and satisfies $K^{\\!T}=-K$.\n\nWrite\n\\[\n   I_n-Q=\n   \\begin{bmatrix}\n         I_d-A & -B\\\\\n         -C    & I_{\\,n-d}-D\n   \\end{bmatrix},\\qquad\n   I_n+Q=\n   \\begin{bmatrix}\n         I_d+A &  B\\\\\n         C     & I_{\\,n-d}+D\n   \\end{bmatrix}.\n\\]\nBecause $I_{\\,n-d}-D=R^{-1}$ is invertible, the inverse of $I_n-Q$\ncan be expressed with its Schur complement\n\\[\n      E:=I_d-A-BR\\,C .\n\\tag{4}\n\\]\nA standard block–matrix inversion yields\n\\[\n   (I_n-Q)^{-1}=\n   \\begin{bmatrix}\n      E^{-1} & E^{-1}B R\\\\\n      -R C E^{-1} & R-R C E^{-1}B R\n   \\end{bmatrix}.\n\\]\nMultiplying this inverse by $I_n+Q$ gives\n\\[\n      K=\n      \\begin{bmatrix}\n          E^{-1}(I_d+A+BR\\,C) & *\\\\\n          *                   & *\n      \\end{bmatrix}.\n\\tag{5}\n\\]\nIn other words,\n\\[\n      K_{11}=E^{-1}S=:J .\n\\tag{6}\n\\]\n\nSince $K^{\\!T}=-K$, each diagonal block of $K$ is itself\nskew–symmetric; hence\n\\[\n      J^{\\!T}=-J .\n\\tag{7}\n\\]\n\n\\medskip\n\\underline{Singularity of $S$.}  \nBecause $E$ is a Schur complement of $I_n-Q$, formula (4) shows\n$E$ is invertible.  From (6) we have\n$\\det S=\\det(E)\\det J$.  \nNow $d$ is odd and $J$ is skew–symmetric, so\n$\\det J=0$; hence $\\det S=0$ and $S$ is singular.\n\n\\smallskip\n\\emph{(ii)  Construction of $+1$–eigenvectors of $T$.}  \n\nRelative to $W\\oplus W^{\\perp}$ we have\n\\[\n      P=\\operatorname{diag}\\!\\bigl(-I_{d},\\,I_{\\,n-d}\\bigr),\\qquad\n      T=QP=\n      \\begin{bmatrix}\n          -A & B\\\\\n          -C & D\n      \\end{bmatrix}.\n\\tag{8}\n\\]\n\nPick any non–zero $x\\in\\ker S$ and set\n\\[\n      y:=-R\\,C\\,x .\n\\tag{9}\n\\]\nDefine $w(x)$ as in the statement, i.e. $w(x)=\\bigl[x^{\\!T},\\,y^{\\!T}\\bigr]^{\\!T}$.  \nUsing $y=-R\\,C\\,x$ and the identity $DR=R-I_{\\,n-d}$ we compute\n\\[\n   Tw(x)=\n   \\begin{bmatrix}\n          -A & B\\\\ -C & D\n   \\end{bmatrix}\n   \\begin{bmatrix}\n          x\\\\ y\n   \\end{bmatrix}\n   =\n   \\begin{bmatrix}\n          -A x + B y\\\\[2pt]\n          -C x + D y\n   \\end{bmatrix}.\n\\]\nThe lower block equals\n\\[\n    -C x + D y\n    = -C x - D R C x\n    = -C x - (I_{\\,n-d}-R) C x\n    = -C x - C x + R C x\n    = y ,\n\\]\nwhile the upper block simplifies to\n\\[\n     -A x + B y\n     = -A x - B R C x\n     = -\\bigl(A + B R C + I_d\\bigr)x + x\n     = x ,\n\\]\nbecause $x\\in\\ker S$ precisely means $(I_d+A+BR\\,C)x=0$.  Hence\n$Tw(x)=w(x)$.\n\nInjectivity of $x\\mapsto w(x)$ implies $\\dim\\ker(I_n-T)\\ge\\dim\\ker S\\ge 1$.\nRepeating the parity argument of the draft (unchanged by the present\nsign corrections) shows that the algebraic multiplicity $r$ of the\neigenvalue $1$ in $T$ satisfies $r\\equiv d\\pmod 2$.\n\n\\vspace{6pt}\n\\textbf{(d)  A constructive Cartan–Dieudonné factorisation.}\n\n\\emph{Algorithm.}  \nLet $Q_0:=Q$ and for $k=1,\\dots,n$ do:\n\n\\smallskip\n\\emph{Step $k$.}  \nIf $Q_{k-1}e_k=e_k$, do nothing;  \notherwise put\n\\[\n      u_k:=Q_{k-1}e_k-e_k\\neq 0,\\qquad\n      P_k:=I_n-\\frac{2\\,u_k u_k^{\\!T}}{u_k^{\\!T}u_k},\\qquad\n      Q_k:=P_kQ_{k-1}.\n\\]\nThen $Q_ke_k=e_k$.  After step $k$ the first $k$ columns of $Q_k$\ncoincide with those of $I_n$; consequently $Q_n=I_n$ and\n\\[\n      Q=P_1P_2\\cdots P_s ,\n\\tag{10}\n\\]\nwhere $s\\le n$ is the number of non–trivial steps.\nEach $P_k$ is a genuine rank–one Householder reflection, so at most\n$n$ such reflections are required.\n\n\\smallskip\n\\emph{Sharpness for $-I_n$.}  \nAssume $-I_n=R_1\\cdots R_m$ with rank–one reflections $R_j$.\nSince $\\det(-I_n)=(-1)^n$ and $\\det R_j=-1$, parity forces\n$m\\equiv n\\pmod 2$.  \nIf $m<n$, the $(n-1)$–dimensional fixed hyperplanes of the $R_j$\nwould have a non–trivial intersection, giving a non–zero vector fixed\nby $-I_n$, a contradiction.  Thus $m\\ge n$, and the bound $n$ in (10)\nis best possible.\n\n\\vspace{6pt}\n\\textbf{(e)  Minimal numbers inside $SO(n)$.}\n\n\\emph{Upper bounds.}  \nFactorisation (10) works for every $Q\\in SO(n)$ and uses at most\n$n$ reflections.  Because $\\det P_k=-1$ for each reflection,\nthe total number of reflections produced by the algorithm is\n\\emph{even}.  Hence\n\\[\n      r(n)\\le\n      \\begin{cases}\n            n,   & n\\ \\text{even},\\\\[6pt]\n            n-1, & n\\ \\text{odd}.\n      \\end{cases}\n\\tag{11}\n\\]\n\n\\emph{Lower bounds, $n$ even.}  \nFor $n=2m$ put\n\\[\n     Q_{\\mathrm{even}}\n     :=\\operatorname{diag}\\bigl(R(\\pi),\\dots,R(\\pi)\\bigr)\\in SO(n),\n     \\qquad\n     R(\\pi)=\n     \\begin{bmatrix}\n        -1&0\\\\ 0&-1\n     \\end{bmatrix}.\n\\]\nThus $Q_{\\mathrm{even}}=-I_n$, which part (d) showed requires exactly\n$n$ reflections, so $r(2m)\\ge 2m$.\n\n\\emph{Lower bounds, $n$ odd.}  \nFor $n=2m+1$ take\n\\[\n     Q_{\\mathrm{odd}}\n     :=\\operatorname{diag}\\bigl(R(\\pi),\\dots,R(\\pi),1\\bigr)\\in SO(n).\n\\]\nHere $Q_{\\mathrm{odd}}$ acts as $-I$ on an $(n-1)$–dimensional\nsubspace.  A product of fewer than $n-1$ reflections fixes at least a\n$2$–dimensional subspace, so it cannot be $Q_{\\mathrm{odd}}$.\nTherefore $r(2m+1)\\ge 2m$.\n\nCombining the bounds with (11) gives\n\\[\n      r(n)=\n      \\begin{cases}\n            n,   & n\\ \\text{even},\\\\[6pt]\n            n-1, & n\\ \\text{odd}.\n      \\end{cases}\n      \\qquad\\qquad\\square\n\\]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.655667",
        "was_fixed": false,
        "difficulty_analysis": "• Higher–dimensional reflection:  The problem replaces a single\n  hyperplane reflection (d = 1) by a reflection through an arbitrary\n  odd-dimensional subspace, forcing the solver to cope with matrices of the\n  form I − 2U(UᵀU)⁻¹Uᵀ and to control their action on two complementary\n  subspaces simultaneously.\n\n• Quantitative bounds:  Part (b) does not merely assert the existence of\n  an eigenvalue 1; it demands a lower bound on its multiplicity, requiring\n  rank arguments that intertwine Q and P in a non-trivial way.\n\n• Interaction with the −1–eigenspace:  Part (c) brings in κ₋, the\n  multiplicity of −1 for Q, and asks for a sharp estimate.  The solver must\n  combine spectral information on Q with structural properties of P.\n\n• Factorisation problem:  Part (d) pushes the classical “every orthogonal\n  matrix is a product of reflections’’ into an optimisation question, whose\n  solution needs an iterative, dimension-counting construction.\n\n• Orientation constraint and SO(n):  Part (e) introduces parity issues for\n  determinants and forces a delicate analysis to obtain the exact number of\n  Householder reflections needed inside SO(n).\n\nAltogether these additions demand deeper linear-algebraic insight,\nspectral analysis, and combinatorial reasoning far beyond the original\nkernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}