On the Relationship between Conditional (CAR) and Simultaneous
(SAR) Autoregressive Models
Jay M. Ver Hoefa, Ephraim M. Hanksb, Mevin B. Hootenc
a Marine Mammal Laboratory, NOAA Alaska Fisheries Science Center,
7600 Sand Point Way NE, Seattle, WA 98115, tel: (907) 456-1995
and
b Department of Statistics, The Pennsylvania State University
and
c U.S. Geological Survey, Colorado Cooperative Fish and Wildlife Research Unit,
Department of Fish, Wildlife, and Conservation Biology,
and Department of Statistics, Colorado State University
October 20, 2017
arXiv:1710.07000v1  [math.ST]  19 Oct 2017

Abstract
We clarify relationships between conditional (CAR) and simultaneous (SAR) autoregressive mod-
els. We review the literature on this topic and ﬁnd that it is mostly incomplete. Our main result is
that a SAR model can be written as a unique CAR model, and while a CAR model can be written
as a SAR model, it is not unique. In fact, we show how any multivariate Gaussian distribution
on a ﬁnite set of points with a positive-deﬁnite covariance matrix can be written as either a CAR
or a SAR model. We illustrate how to obtain any number of SAR covariance matrices from a
single CAR covariance matrix by using Givens rotation matrices on a simulated example. We also
discuss sparseness in the original CAR construction, and for the resulting SAR weights matrix.
For a real example, we use crime data in 49 neighborhoods from Columbus, Ohio, and show that
a geostatistical model optimizes the likelihood much better than typical ﬁrst-order CAR models.
We then use the implied weights from the geostatistical model to estimate CAR model parameters
that provides the best overall optimization.
Key Words: lattice models; areal models, spatial statistics, covariance matrix

1
Introduction
Cressie (1993, p. 8) divides statistical models for data collected at spatial locations into two broad
classes: 1) geostatistical models with continuous spatial support, and 2) lattice models, also called
areal models (Banerjee et al., 2004), where data occur on a (possibly irregular) grid, or lattice, with
a countable set of nodes or locations. The two most common lattice models are the conditional
autoregressive (CAR) and simultaneous autoregressive (SAR) models, both notable for sparseness
of their precision matrices. These autoregressive models are ubiquitous in many ﬁelds, including
disease mapping (e.g., Clayton and Kaldor, 1987; Lawson, 2013), agriculture (Cullis and Gleeson,
1991; Besag and Higdon, 1999), econometrics (Anselin, 1988; LeSage and Pace, 2009), ecology
(Lichstein et al., 2002; Kissling and Carl, 2008), and image analysis (Besag, 1986; Li, 2009). CAR
models form the basis for Gaussian Markov random ﬁelds (Rue and Held, 2005) and the popular
integrated nested Laplace approximation methods (INLA, Rue et al., 2009), and SAR models are
popular in geographic information systems (GIS) with the GeoDa software (Anselin et al., 2006).
Hence, both CAR and SAR models serve as the basis for countless scientiﬁc conclusions. Because
these are the two most common classes of models for lattice data, it is natural to compare and
contrast them. There has been sporadic interest in studying the relationships between CAR and
SAR models (e.g., Wall, 2004), and how one model might or might not be expressed in terms of
the other (Haining, 1990; Cressie, 1993; Martin, 1987; Waller and Gotway, 2004), but there is little
clarity in the existing literature on the relationships between these two classes of autoregressive
models.
Our goal is to clarify, and add to, the existing literature on the relationships between CAR
and SAR covariance matrices, by showing that any positive-deﬁnite covariance matrix for a mul-
tivariate Gaussian distribution on a ﬁnite set of points can be written as either a CAR or a SAR
covariance matrix, and hence any valid SAR covariance matrix can be expressed as a valid CAR
covariance matrix, and vice versa.
This result shows that on a ﬁnite dimensional space, both
SAR and CAR models are completely general models for spatial covariance, able to capture any
positive-deﬁnite covariance. While CAR and SAR models are among the most commonly-used
spatial statistical models, this correspondence between them, and the generality of both models,
has not been fully described before now. These results also shed light on some previous literature.
This paper is organized as follows: In Section 2, we review SAR and CAR models and lay
out necessary conditions for these models. In Section 3, we provide theorems that show how to
obtain SAR and CAR covariance matrices from any positive deﬁnite covariance matrix, which also
establishes the relationship between CAR and SAR covariance matrices. In Section 4, we provide
examples of obtaining SAR covariance matrices from a CAR covariance matrix on fabricated data,
and a real example for obtaining a CAR covariance matrix for a geostatistical covariance matrix.
Finally, in Section 5, we conclude with a detailed discussion of the incomplete results of previous
literature.
2
Review of SAR and CAR models
In what follows, we denote matrices with bold capital letters, and their ith row and jth column
with small case letters with subscripts i, j; for example, the i, jth element of C is ci,j. Vectors are
denoted as lower case bold letters. Let Z ≡(Z1, Z2, . . . , Zn)T be a vector of n random variables
at the nodes of a graph (or junctions of a lattice). The edges in the graph, or connections in the
1

lattice, deﬁne neighbors, which are used to model spatial dependency.
2.1
SAR Models
Consider the SAR model with mean zero. An explicit autocorrelation structure is imposed,
Z = BZ + ν,
(1)
where the n × n spatial dependence matrix, B, is relating Z to itself, and ν ∼N(0, Ω), where Ω
is diagonal with positive values. These models are generally attributed to Whittle (1954). Solving
for Z, note that sites cannot depend on themselves so B will have zeros on the diagonal, and that
(I −B)−1 must exist (Cressie, 1993; Waller and Gotway, 2004), where I is the identity matrix.
Then Z ∼N(0, ΣSAR), where
ΣSAR = (I −B)−1Ω(I −BT )−1;
(2)
see, for example, Cressie (1993, p. 409). The spatial dependence in the SAR model is due to the
matrix B which causes the simultaneous autoregression of each random variable on its neighbors.
Note that B does not have to be symmetric because it does not appear directly in the inverse of
the covariance matrix (i.e., precision matrix). The covariance matrix must be positive deﬁnite.
For SAR models, it is enough that (I −B) is nonsingular (i.e., that (I −B)−1 exists), because the
quadratic form, writing it as (I −B)−1Ω[(I −B)−1]T , with Ωcontaining positive diagonal values,
ensures ΣSAR will be positive deﬁnite.
In summary, the following conditions must be met for ΣSAR in (2) to be a valid SAR
covariance matrix:
S1
(I −B) is nonsingular,
S2
Ωis diagonal with positive elements, and
S3
bi,i = 0, ∀i.
2.2
CAR models
The term “conditional,” in the CAR model, is used because each element of the random process
is speciﬁed conditionally on the values of the neighboring nodes. Let Zi be a random variable at
the ith location, again assuming that the expectation of Zi is zero for simplicity, and let zj be its
realized value. The CAR model is typically speciﬁed as
Zi|z−i ∼N

X
∀ci,j̸=0
ci,jzj, mi,i

,
(3)
where z−i is the vector of all zj where j ̸= i, C is the spatial dependence matrix with ci,j as its
i, jth element, ci,i = 0, and M is a diagonal matrix with positive diagonal elements mi,i. Note that
mi,i may depend on the values in the ith row of C. In this parameterization, the conditional mean
of each Zi is weighted by values at neighboring nodes. The variance component, mi,i, often varies
with node i, and thus M is generally nonstationary. In contrast to SAR models, it is not obvious
that (3) leads to a full joint distribution for Z. Besag (1974) used Brook’s lemma (Brook, 1964) and
2

the Hammersley-Cliﬀord theorem (Hammersley and Cliﬀord, 1971; Cliﬀord, 1990) to show that,
when (I −C)−1M is positive deﬁnite, Z ∼N(0, ΣCAR), with
ΣCAR = (I −C)−1M.
(4)
ΣCAR must be symmetric, requiring
ci,j
mi,i
= cj,i
mj,j
, ∀i, j.
(5)
Most authors describe CAR models as the construction (3), with condition that ΣCAR must be
positive deﬁnite given the symmetry condition (5). However, a more speciﬁc statement is possible
on the necessary conditions for (I −C), making a comparable condition to S1 for SAR models. We
provide a novel proof, Proposition 3 in the Appendix, showing that if M is positive deﬁnite along
with (5) (forcing symmetry on ΣCAR), it is only necessary for (I −C) to have positive eigenvalues
for ΣCAR to be positive deﬁnite.
In summary, the following conditions must be met for ΣCAR in (4) to be a valid CAR
covariance matrix:
C1
(I −C) has positive eigenvalues,
C2
M is diagonal with positive elements,
C3
ci,i = 0, ∀i, and
C4
ci,j/mi,i = cj,i/mj,j, ∀i, j.
2.3
Weights Matrices
In practice, B = ρsW and C = ρcW are usually used to construct valid SAR and CAR models,
where W is a weights matrix with wi,j ̸= 0 when locations i and j are neighbors, otherwise
wi,j = 0. Neighbors are typically pre-speciﬁed by the modeler. When i and j are neighbors, we
often set wi,j = 1, or use row-standardization so that Pn
j=1 wi,j = 1; that is, dividing each row
in unstandardized W by wi,+ ≡Pn
j=1 wi,j yields an asymmetric row-standardized matrix that we
denote as W+. For CAR models, deﬁne M+ as the diagonal matrix with mi,i = 1/wi,+, then (5)
is satisﬁed. The row-standardized CAR model can be written equivalently as
Σ+ = σ2(I −ρcW+)−1M+ = σ2(diag(W1) −ρcW)−1,
(6)
where 1 is a vector of all ones, σ2 is an overall variance parameter, and diag(·) creates a diagonal
matrix from a vector. A special case of the CAR model, called the intrinsic autoregressive model
(IAR) (Besag and Kooperberg, 1995), occurs when ρc = 1, but the covariance matrix does not
exist, so we do not consider it further.
There can be confusion on how ρ is constrained for SAR and CAR models, which we now
clarify. Suppose that W has all real eigenvalues. Let {λi} be the set of eigenvalues of W, and
let {ωi} be the set of eigenvalues of (I −ρW). Then, in the Appendix (Proposition 4), we show
that ωi = (1 −ρλi). First, notice that if λi = 0, then ωi = 1 for all ρ. Hence, (I −ρW) will be
nonsingular for all ρ /∈{λ−1
i } whenever λi ̸= 0, which is suﬃcient for SAR model condition S1.
Note that it is possible for all ωi to be positive, even when ρW has some zero eigenvalues (λi = 0),
and thus our result is more general than that of Li et al. (2007), who only consider the case when
3

all λi ̸= 0. If any λi ̸= 0, then at least two λi are nonzero because tr(W) = Pn
i=1 λi = 0. If
at least two eigenvalues are nonzero, then λ[1], the smallest eigenvalue of W, must be less than
zero, and λ[N], the largest eigenvalue of W, must be greater than zero. Then 1/λ[1] < ρ < 1/λ[N]
ensures that (I−ρW) has positive eigenvalues (Appendix, Proposition 4) and satisﬁes condition C1
for CAR models. For SAR models, if (I −ρW) has positive eigenvalues it is also nonsingular, so
1/λ[1] < ρ < 1/λ[N] provides a suﬃcient (but not necessary) condition for condition S1.
In practice, the restriction 1/λ[1] < ρ < 1/λ[N] is often used for both CAR and SAR
models. When considering W+, the restriction becomes 1/λ[1] < ρ < 1, where usually 1/λ[1] < −1.
Wall (2004) shows irregularities for negative ρ values near the lower bound for both SAR and
CAR models, thus many modelers simply use −1 < ρ < 1. In fact, in many cases, only positive
autocorrelation is expected, so a further restriction is used where 0 < ρ < 1 (e.g., Li et al., 2007).
For these constructions, ρ typically has more positive marginal autocorrelation with increasing
positive ρ values, and more negative marginal autocorrelation with decreasing negative ρ values
(Wall, 2004). There has been little research on the behavior of ρ outside of these limits for SAR
models.
Our goal is to develop relationships that allow a CAR covariance matrix, satisfying conditions
C1 - C4, to be obtained from a SAR covariance matrix, satisfying conditions S1 - S3, and vice versa.
We develop these in the next section, and, in the Discussion and Conclusions section, we contrast
our results to the incomplete results of previous literature.
3
Relationships between CAR and SAR models
Assume a covariance matrix for a SAR model as given in (2), and a covariance matrix for a CAR
model as given in (4). We show that any zero-mean Gaussian distribution on a ﬁnite set of points,
Z ∼N(0, Σ), can be written with a covariance matrix parameterized either as a CAR model,
Σ = (I −C)−1M, or as a SAR model, Σ = (I −B)−1Ω(I −BT )−1.
It is straightforward to
generalize to the case where the mean is nonzero so, for simplicity of notation, we use the zero
mean case. A corollary is that any CAR covariance matrix can be written as a SAR covariance
matrix, and vice versa. Before proving the theorems, some preliminary results are useful.
Proposition 1. If D is a square diagonal matrix, and Q is a square matrix with zeros on the
diagonal, then, provided the matrices are conformable, both DQ and QD have zeros on the diagonal.
Proof. We omit the proof because it is apparent from the algebra of matrix products.
Proposition 2. Let A, B, and C be square matrices. If A = BC, and A and C have inverses,
then B has a unique inverse.
Proof. Because C has an inverse, B = AC−1, and because A has an inverse, B−1 = CA−1. B−1
is unique because it is square and full-rank (e.g., Harville, 1997, p. 80).
We now prove that both SAR and CAR covariance matrices are suﬃciently general to
represent any ﬁnite-dimensional positive-deﬁnite covariance matrix.
Theorem 1. Any positive deﬁnite covariance matrix Σ can be expressed as the covariance matrix
of a SAR model (I −B)−1Ω(I −BT )−1, (2), for a (non-unique) pair of matrices B and Ω.
4

Proof. We consider a constructive proof and show that the matrices B and Ωsatisfy conditions S1
- S3.
(i) Write Σ−1 = LLT , and suppose that L is full rank with positive eigenvalues. Note that
L is not unique.
A Cholesky decomposition could be used, where L is lower triangular,
or a spectral (eigen) decomposition could be used, where Σ = VEVT , with V containing
orthonormal eigenvectors and E containing eigenvalues on the diagonal and zeros elsewhere.
Then L = VE−1/2, where the diagonal matrix E−1/2 contains reciprocals of square roots of
the eigenvalues in E.
(ii) Decompose L into L = G −P where G is diagonal and P has zeros on the diagonal. Then
LLT = (G −P)(GT −PT ) by construction.
(iii) Then set
Ω−1 = GG and BT = PG−1.
(7)
Note that because L has positive eigenvalues, then ℓi,i > 0, and because G is diagonal with
gi,i = ℓi,i, G−1 exists.
Then Σ−1 = (I −BT )Ω−1(I −B), expressed in SAR form (2). The matrices B and Ωsatisfy S1 -
S3, as follows.
(S1) Note that P = BT G, so L = G −P = (I −BT )G and LT = G(I −B). Then, by Proposition
2, (I −B)−1 and (I −BT )−1 exist.
(S2) Because G is diagonal, Ωis diagonal with ωi,i = g2
i,i > 0.
(S3) By Proposition 1, bi,i = 0 because BT = PG−1.
Theorem 2. Any positive-deﬁnite covariance matrix Σ can be expressed as the covariance matrix
of a CAR model (I −C)−1M, (4), for a unique pair of matrices C and M (Cressie, 1993, p. 434).
Proof. We add an explicit, constructive proof of the result given by Cressie (1993, p. 434) by
showing that matrices C and M are unique and satisfy conditions C1 - C4.
(i) Let Q = Σ−1 and decompose it into Q = D−R, where D is diagonal with elements di,i = qi,i
(the diagonal elements of the precision matrix Q), and R has zeros on the diagonal (ri,i = 0)
and oﬀ-diagonals equal to ri,j = −qi,j.
(ii) Set
C = D−1R and M = D−1.
(8)
Then Σ−1 = D −R = D(I −D−1R) = M−1(I −C), with Σ expressed in CAR form (4). The
matrices C and M, from (8), are uniquely determined by Σ because Σ and D have unique inverses,
and satisfy C1 - C4, as follows.
(C1) M is strictly diagonal with positive values, so M and M−1 are positive deﬁnite. By hypothesis,
Σ, and hence Σ−1 are positive deﬁnite. Then Σ−1M = (I −C), so by Proposition 3 in the
Appendix, (I −C) has positive eigenvalues.
5

(C2) mi,i = 1/qi,i, and because Q = Σ−1 is positive deﬁnite, we have that qi,i > 0, i = 1, 2, . . . , n.
Thus, each mi,i > 0. By construction, mi,j = 0 for i ̸= j.
(C3) By Proposition 2.1, ci,i = 0 because C = D−1R.
(C4) For i ̸= j, we have that ci,j = d−1
i,i ri,j. As mi,i = d−1
i,i = qi,i, we have that
ci,j
mi,i
=
d−1
i,i ri,j
d−1
i,i
= ri,j = −qi,j.
Because Q = Σ−1 is symmetric, qi,j = qj,i and ci,j/mi,i = cj,i/mj,j.
Having shown that any positive deﬁnite matrix Σ can be expressed as either the covariance
matrix of a CAR model or the covariance matrix of a SAR model, we have the following corollary.
Corollary 1. Any SAR model can be written as a unique CAR model, and any CAR model can be
written as a non-unique SAR model.
Proof. The proof follows directly by ﬁrst noting that a SAR model yields a positive-deﬁnite covari-
ance matrix, and applying Theorem 2, and then noting that a CAR model yields a positive-deﬁnite
covariance matrix, and applying Theorem 1.
The following corollary gives more details on the non-unique nature of the SAR models.
Corollary 2. Any positive-deﬁnite covariance matrix can be expressed as one of an inﬁnite number
of B matrices that deﬁne the SAR covariance matrix in (2).
Proof. Write Σ−1 = LLT as in Theorem 1. Let Ah,s(θ) be a Givens rotation matrix (Golub and
Van Loan, 2013), which is a sparse orthonormal matrix that rotates angle θ through the plane
spanned by the h and s axes. The elements of Ah,s(θ) are as follows. For i /∈{h, s}, ai,i = 1. For
i ∈{h, s}, ah,h = as,s = cos(θ), ah,s = sin(θ) and as,h = −sin(θ). All other entries of Ah,s(θ) are
equal to zero. Notice that Σ−1 = LLT = L(AT
h,s(θ)Ah,s(θ))LT = L∗LT
∗, where L∗= LAT
h,s(θ). A
SAR covariance matrix can be developed as readily for L∗as for L in the proof of Theorem 1. Any
of the inﬁnite values of θ ∈[0, 2π) will result in a unique Ah,s(θ), leading to a diﬀerent L∗, and a
diﬀerent B matrix in (7), but yielding the same positive-deﬁnite covariance matrix Σ.
3.1
Implications of Theorems and Corollaries
Note that for Corollary 2, additional B matrices that deﬁne a ﬁxed positive-deﬁnite covariance
matrix in Corollary 2 could also be obtained by repeated Givens rotations.
For example, let
L∗= LAT
1,2(θ)AT
3,4(η) for angles θ and η. Then a new B can be developed for this L∗just as
readily as those in the proof to Corollary 2. We use this idea extensively in the examples.
Theorem 1 helps clarify the use of Ω. Authors often write the SAR model as (I −B)−1(I −
BT )−1, assuming that Ω= I in (2). In the proofs to Theorem 1 and Corollary 2, this requires
ﬁnding L with ones on the diagonal so that G = I. It is interesting to consider if one can always
ﬁnd such L, which would justify the practice of using the simpler form, (I −B)−1(I −BT )−1, for
SAR models. We leave that as an open question.
6

G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
1
2
3
4
5
1
2
3
4
5
Column
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
(a)
5
10
15
20
25
25
20
15
10
5
Column (Location Index)
Row (Location Index)
(b)
5
10
15
20
25
25
20
15
10
5
Column (Location Index)
Row (Location Index)
(c)
5
10
15
20
25
25
20
15
10
5
Column (Location Index)
Row (Location Index)
(d)
G
GGGGGGG
GGGGGG
GGG
GGGGGG
GGGGGG
GGGGG
G
GGGG
GGGGG
G
GGGGGGGGGGGGG
GGGGGGGGGGGGG
G
GGGGGGGG
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
GGGGGGGGGGGGG
GGGGGGGGGGGGGGGGGGGGGGGGG
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
0
500
1000
1500
2000
9.0
9.5
10.0
10.5
Iteration
Fullness Value
0 to 0.001
0.001 to 0.01
0.01 to 0.1
0.1 to 0.2
0.2 to 0.4
0.4 to 0.8
0.8 to 1
Matrix Weights
(e)
5
10
15
20
25
25
20
15
10
5
Column (Location Index)
Row (Location Index)
(f)
Figure 1: Sparseness in CAR and SAR models. (a)
5 × 5 grid of spatial locations, with lines connecting
neighboring sites. The numbers in the circles are in-
dexes of the locations. (b) Graphical representation
of weights in the ρW+ matrix in the CAR model.
The color legend is given below. (c) Graphical rep-
resentation of weights in the B matrix when using the
Cholesky decomposition, and (d) when using spectral
decomposition. (e) Fullness function during minimiza-
tion when searching for sparseness. (f) Graphical rep-
resentation of weights in the B matrix at the termi-
nation of an algorithm to search for sparseness using
Givens rotations on the spectral decomposition in (d).
In Section 2.3, we discussed how most
CAR and SAR models are constructed by con-
straining ρ in ρW. Consider Theorem 1, where
L is a lower-triangular Cholesky decomposi-
tion. Then P has zero diagonals and is strictly
lower triangular, and so BT = PG−1 is strictly
lower triangular.
In this construction, all of
the eigenvalues of B are zero. Thus, for SAR
models, there are unexplored classes of models
that do not depend on the typical construction
B = ρW.
Most CAR and SAR models are devel-
oped such that C and B are sparse matrices,
containing mostly zeros, but containing posi-
tive elements whose weights depend locally on
neighbors. Although we demonstrated how to
obtain a CAR covariance matrix from a SAR
covariance matrix, and vice versa, there is no
guarantee that using a sparse C in a CAR
model will yield a sparse B in a SAR model,
or vice versa. We explore this idea further in
the following examples.
4
Examples
We provide two examples, one where we illus-
trate Theorem 1 primarily, and a second where
we use Theorem 2. In the ﬁrst, we fabricated
a simple neighborhood structure and created a
positive deﬁnite matrix by a CAR construction.
Using Givens rotation matrices, we then ob-
tained various non-unique SAR covariance ma-
trices from the CAR covariance matrix. We also
explore sparseness in B for SAR models when
they are obtained from sparse C for CAR mod-
els.
For a second example, we used real data
on neighborhood crimes in Columbus, Ohio.
We model the data with the two most common
CAR models, using a ﬁrst-order neighborhood
model where C is both unstandardized and row-
standardized. Then, from a positive-deﬁnite co-
variance matrix obtained from a geostatistical
model, we obtain the equivalent and unique CAR covariance matrix. We use the weights obtained
from the geostatistical covariance matrix to allow further CAR modeling, ﬁnding a better likelihood
7

optimization than both the unstandardized and row-standardized ﬁrst-order CAR models.
Consider the graph in Figure 1a, which shows an example of neighbors for a CAR model.
Using one to indicate a neighbor, and zeros elsewhere, the W matrix was used to create the
row-standardized W+ matrix in (6). Values of ρcW+, where ρc = 0.9, are shown graphically in
Figure 1b. For the resulting covariance matrix, Σ+ in (6), the Cholesky decomposition was used
to create L as in Theorem 1. Using (7) in Theorem 1, the weights matrix B created from L is
shown in Figure 1c. For the same covariance matrix Σ+, we also used the spectral decomposition
to create L as in Theorem 1. The weights matrix B created from this L, using (7) in Theorem 1,
is shown in Figure 1d. Note that the B matrix in Figure 1d is less sparse than B in Figure 1c,
although they both yield exactly the same covariance matrix by the SAR construction (2), which
we veriﬁed numerically. Figure 1c also veriﬁes our comments in Section 3.1; that there exists some
B where all eigenvalues are zero (because all diagonal elements are zero).
We also sought to transform the B matrix in Figure 1d to a sparser form using the proof to
Corollary 2 and the Given’s rotations. For a vector x of length n, an index of sparseness (Hoyer,
2004) is
sparseness(x) =
√n −
P
i |xi|
√P
i x2
i
√n −1
,
which ranges from zero to one. Ignoring the dimensions of a matrix, we create the matrix function
f(B) =
P
i,j |bi,j|
qP
i,j b2
i,j
,
which is a measure of the fullness of a matrix. We propose an iterative algorithm to minimize
f(B) for orthonormal Givens rotations as explained in Corollary 2. Let Lh,s(θ) = LAT
h,s(θ), where
L = VE−1/2V−1 used the spectral decomposition of Σ+ as in the proof of Theorem 1, and Ah,s(θ)
is a Givens rotation matrix as in the proof of Corollary 2. Denote θ∗
k as the value of θ that minimizes
f(B) when B is created by decomposing LAT
h,s(θ) into P and G (as in (ii) in Theorem 1), while
constraining θ to values satisfying bi,j ≥0 ∀i, j. Then L[1]
1,2 ≡LAT
1,2(θ∗
1), where k = 1 is the ﬁrst
iteration. For the second iteration, let θ∗
2 be the value that minimizes f(B) for B created from
L[1]
1,2AT
1,3(θ), and hence for k = 2, L[2]
1,3 ≡L[1]
1,2AT
1,3(θ∗
2). We cycled through h = 1, 2, . . . , 24 and
s = (h + 1), . . . , 25 for each iteration k in a coordinate decent minimization of f(B). We cycled
through all of h and s eight times for a total of 8(25)(25 −1)/2 = 2400 iterations. The value
of f(B) for each iteration is plotted in Figure 1e and the ﬁnal B matrix is given in Figure 1f.
Although we did not achieve the sparsity of Figure 1c, we were able to increase sparseness from
the starting matrix in Figure 1d. Note that the B matrix depicted in Figure 1f yields exactly the
same covariance matrix as the B matrices shown in Figures 1c,d. There are undoubtedly better
ways to minimize f(B), such as simulated annealing (Kirkpatrick et al., 1983), and there may be
alternative optimization criteria. We do not pursue these here. Our goal was to show that it is
possible to explore many conﬁgurations of matrix weights in SAR models, which produce equivalent
covariance matrices, by using orthonormal Givens rotations of the L matrix.
8

4.1
Columbus Crime Data
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G G
G
G
G
G
G
G
G
G
G
G
G G
G
G
G
G
G
G
G
G
G
G G
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
under 20.05
20.05 − 34
34 − 48.59
over 48.59
Range
Figure 2:
Columbus crime map, in rates per 1000
people. Numbers in each polygon are the indexes for
locations, and the white lines show ﬁrst-order neigh-
bors. The estimated range parameter from the spher-
ical geostatistical model is shown at the bottom.
The Columbus data are found in the spdep
package (Bivand et al., 2013; Bivand and Pi-
ras, 2015) for R (R Core Team, 2016). Figure 2
shows 49 neighborhoods in Columbus, Ohio.
We used residential burglaries and vehicle thefts
per thousand households in the neighborhood
(Anselin, 1988, Table 12.1, p. 189) as the re-
sponse variable. Spatial pattern among neigh-
borhoods appeared autocorrelated (Figure 2),
with higher crime rates in the more central
neighborhoods. When analyzing rate data, it is
customary to account for population size (e.g.,
Clayton and Kaldor, 1987), which aﬀects the
variance of the rates. However, for illustrative
purposes, we used raw rates. A histogram of
the data appeared approximately bell-shaped,
thus we assumed a Gaussian distribution with
a covariance matrix containing autocorrelation
among locations.
First-order neighbors were also taken
from the spdep package for R, and are shown
by white lines in Figure 2. Using a one to indi-
cate a neighbor, and zero otherwise, we denote
the 49 × 49 matrix of weights as Wun, and the
CAR precision matrix has C = ρunWun and
M = σ2
unI in (4). Using the eigenvalues of Wun, the bounds for ρun were -0.335 < ρun < 0.167. We
added a constant independent diagonal component, δ2
unI (also called the nugget eﬀect in geostatis-
tics), so the covariance matrix was Σun = σ2
un(I −ρunWun)−1 + δ2
unI. Denote the crime rates as
y. We assumed a constant mean, so y ∼N(1µ, Σun), where 1 is a vector of all ones. Let L(θun|y)
be minus 2 times the restricted maximum likelihood equation (REML, Patterson and Thompson,
1971, 1974) for the crime data, where the set of covariance parameters is θun = (σ2
un, ρun, δ2
un)T . We
optimized the likelihood using REML and obtained L(ˆθun|y) = 388.83. Recall that CAR models
have nonstationary variances and covariances (e.g., Wall, 2004). The marginal variances of the
estimated model are shown in Figure 3a, and the marginal correlations are shown in Figure 4a.
9

(a)
under 348
348 − 525
525 − 701
701 − 878
878 − 1055
1055 − 1231
over 1231
(b)
(c)
(d)
Figure 3:
Marginal variances by location.
(a) Unstandardized ﬁrst-order CAR model, (b) Row-
standardized ﬁrst-order CAR model, (c) spherical geostatistical model, (d) CAR model using weights
obtained from geostatistical model.
We also optimized the likelihood using the row-standardized weights matrix, W+ in (6),
which we denote Wrs. In this case, the CAR precision matrix has C = ρrsW+, −1 < ρrs < 1, and
M = σ2
rsM+ in (4). Again we added a nugget eﬀect, so Σrs = σ2
rs(I−ρunW+)−1M+ +δ2
rsI. For the
set of covariance parameters θrs = (σ2
rs, ρrs, δ2
rs)T , we obtained L(ˆθrs|y) = 397.25. This shows that
the unstandardized weights matrix Wun provides a substantially better likelihood optimization
than Wrs. The marginal variances of the row-standardized model are shown in Figure 3b, and
the marginal correlations are shown in Figure 4b. The diﬀerence between L(ˆθun|y) and L(ˆθrs|y)
indicates that the weights matrix C has a substantial eﬀect for these data.
We optimized the likelihood with a geostatistical model next using a spherical autocorrelation
model. Denote the geostatistical correlation matrix as S, where
si,j = [1 −1.5(ei,j/α) + 0.5(ei,j/α)3]I(di,j < α),
and I(•) is the indicator function, equal to one if its argument is true, otherwise it is zero, and ei,j
10

is Euclidean distance between the centroids of the ith and jth polygons in Figure 2. We included
a nugget eﬀect, so Σsp = σ2
spS + δ2
spI. For the set of covariance parameters θsp = (σ2
sp, α, δ2
sp)T ,
we obtained L(ˆθsp|y) = 374.61. The geostatistical model provides a substantially better optimized
likelihood than either the unstandardized or row-standardized CAR model. The marginal variances
of geostatistical models are stationary (Figure 3c). The estimated range parameter, ˆα, is shown by
the lower bar in Figure 2. Any locations separated by a distance greater than that shown by the
bar will have zero correlation (Figure 4c).
under 0.01
0.01 − 0.25
0.25 − 0.5
0.5 − 0.75
0.75 − 0.99
over 0.99
0
10
20
30
40
50
50
40
30
20
10
0
Column (Location Index)
Row (Location Index)
(a)
0
10
20
30
40
50
50
40
30
20
10
0
Column (Location Index)
Row (Location Index)
(b)
0
10
20
30
40
50
50
40
30
20
10
0
Column (Location Index)
Row (Location Index)
(c)
0
10
20
30
40
50
50
40
30
20
10
0
Column (Location Index)
Row (Location Index)
(d)
Figure 4: Marginal correlations, none of which were
below zero. The location indexes are given by the num-
bers in Figure 2. (a) Unstandardized ﬁrst-order CAR
model, (b) Row-standardized ﬁrst-order CAR model,
(c) spherical geostatistical model, (d) CAR model us-
ing weights obtained from geostatistical model.
It appears that the geostatistical model
provides a much better optimized likelihood
than the two most commonly-used CAR mod-
els.
Is it possible to ﬁnd a CAR model to
compete with the geostatistical model?
Us-
ing Theorem 2, we created Ccg and Mcg as
in (8) from the positive deﬁnite covariance ma-
trix from the geostatistical model, Σsp = (I −
ρcgCcg)−1Mcg. Here, we have a CAR represen-
tation that is equivalent to the spherical geosta-
tistical model. Letting Wcg = Ccg, and using
Σcg = σ2
cg(I −ρcgWcg)−1Mcg + δ2
cgI, we opti-
mized for θcg = (σ2
cg, ρcg, δ2
cg)T . For Σcg to be
positive deﬁnite, σ2
cg > 0, -1.104 < ρcg < 1.013,
and δ2
cg ≥0. Because θcg = (1, 1, 0)T is in the
parameter space, we can do no worse than the
spherical geostatistical model. In fact, upon op-
timizing, we obtained L(ˆθcg|y) = 373.95, where
ˆσ2
cg = 0.941, ˆρcg = 1.01, and ˆδ2
cg = 0, a slightly
better optimization than the spherical geosta-
tistical model. The marginal variances for this
geostatistical-assisted CAR model are shown in
Figure 3d, and the marginal correlations are
shown in Figure 4d.
Note the rather large
changes from Figure 3c to Figure 3d, and from
Figure 4c to Figure 4d, with seemingly minor
changes in ˆσ2
cg, from 1 to 0.941, and in ρcg,
from 1 to 1.01. Others have documented rapid
changes in CAR model behavior near the pa-
rameter boundaries, especially for ρcg (Besag
and Kooperberg, 1995; Wall, 2004).
5
Discussion and Conclusions
Haining (1990, p. 89) provided the most comprehensive comparison of the mathematical relation-
ships between CAR and SAR models. He provided several results that we restate using notation
from Sections 2.1 and 2.2, and show that some are incorrect or incomplete.
11

In an attempt to create a CAR covariance matrix from a SAR covariance matrix, assume
that B is a SAR covariance matrix satisfying conditions S1-S3 and Ω= I in (2). Let M = I and C
be symmetric in (4) [which omits the important case (6)]. Then setting SAR and CAR covariances
matrix equal to each other,
(I −C)−1 = [(I −B)(I −BT )]−1 = (I −B −BT + BBT )−1,
(9)
and Haining (1990) claims that C can be obtained from B by setting
C = B + BT −BBT ,
(10)
which is repeated in texts by Waller and Gotway (2004, p. 372) and Schabenberger and Gotway
(2005, p. 339), and in the literature (e.g., Dormann et al., 2007). However, aside from the lack of
generality due to assumptions M = I, Ω= I, and symmetric C, we note that (10) is incomplete
and too limited to be useful, as given in the following remark.
Remark 1. Condition C3 in Section 2.2 is not satisﬁed for C in (10) except when B contains all
zeros.
Proof. Because B has zeros on the diagonal, B + BT will have zeros on the diagonal. Denote bi as
the ith row of B. Then the ith diagonal element of BBT will be the dot product bi • bi, which will
be zero only if all elements of bi are zero. Hence, B + BT −BBT will have zeros on the diagonal
only if B contains all zeros.
In an attempt to create a SAR covariance matrix from a CAR covariance matrix, assume
the same conditions as for (9), and that C is a CAR covariance matrix satisfying conditions C1-C4.
Let (I −C) = SST , where S is a Cholesky decomposition. Haining (1990) suggested S = I −B
and setting B equal to I −S. However, this is incomplete because condition S3 in Section 2.1 will
be satisﬁed only if S has all ones on the diagonal, which also has limited use.
For another approach to relate SAR and CAR covariance matrices, Haining (1990) described
the model F(Z −µ) = Hε, where var(ε) = V. Then E((Z −µ)(Z −µ)T ) = F−1HVHT (F−1)T .
Now let F = (I −C), H = I, and V = (I −C) (this appears to originate in Martin (1987)).
The constructed model is really a SAR model except that it violates condition S2 by allowing
V = (I −C). Alternatively, this can be seen as an attempt to create a SAR model from a CAR
model by assuming an inverse CAR covariance matrix for the error structure of the SAR model,
which gains nothing. Because these arguments are unconvincing, and other authors argue that one
cannot go uniquely from a CAR to a SAR (e.g., Mardia, 1990), we can ﬁnd no further citations
for the arguments of Haining (1990) on obtaining a CAR covariance matrix from a SAR covariance
matrix.
Cressie (1993, p. 409-410) provided a demonstration of how a SAR covariance matrix with
ﬁrst-order neighbors in B leads to a CAR covariance matrix with third-order neighbors in C, and
claims that, generally, there will be no equivalent SAR covariance matrices for ﬁrst and second-
order CAR covariance matrices. However, our demonstration in Figure 1c shows that a sparse B
may be obtained from a sparse CAR model, although it is asymmetric and may not have the usual
neighborhood interpretation.
From Section 2.3, we showed that pre-speciﬁed weights W are often scaled by ρ, and that
ρ is often constrained by the eigenvalues of W. However, we have also discussed in Section 3.1
12

and Figure 1c, that weights can be chosen so that all eigenvalues are zero, for either CAR or SAR
models. We have little information or guidance for developing models where all eigenvalues W are
zero, and this provides an interesting topic for future research.
Wall (2004) provided a detailed comparison on properties of marginal correlation for various
values of ρ when B or C are parameterized as ρsW and ρcW, respectively, but did not develop
mathematical relationships between CAR and SAR models. Lindgren et al. (2011) showed that
approximations to point-referenced geostatistical models based on a ﬁnite element basis expansion
can be expressed as CAR models. In his discussion of the same, Kent (2011) noted that, for a
given geostatistical model of the Matern class, one could construct either a CAR or SAR model
that would approximate the Matern model. This indicates a correspondence between CAR and
SAR models when used as approximations to continuous-space processes, but does not address the
relationship between CAR and SAR models on a native areal support.
Our literature review and discussion showed that there have been scattered eﬀorts to es-
tablish mathematical relationships between CAR and SAR models, and some of the reported re-
lationships are incomplete on the conditions for those relationships. With Theorems 1 and 2 and
Corollary 1, we demonstrated that any zero-mean Gaussian distribution on a ﬁnite set of points,
Z ∼N(0, Σ), with positive-deﬁnite covariance matrix Σ, can be written as either a CAR or a
SAR model, with the important diﬀerence that a CAR model is uniquely determined from Σ but
a SAR model is not so uniquely determined. This equivalence between CAR and SAR models
can also have practical applications. In addition to our examples, the full conditional form of the
CAR model allows for easy and eﬃcient Gibbs sampling (Banerjee et al., 2004, p. 163) and fully
conditional random eﬀects (Banerjee et al., 2004, p. 86). However, spatial econometric models
often employ SAR models (LeSage and Pace, 2009), so easy conversion from SAR to CAR models
may oﬀer computational advantages in hierarchical models and provide insight on the role of fully
conditional random eﬀects. We expect future research will extend our ﬁndings on relationships
between CAR and SAR models and explore novel applications.
Acknowledgments
This research began from a working group on network models at the Statistics and Applied Mathe-
matical Sciences (SAMSI) 2014-15 Program on Mathematical and Statistical Ecology. The project
received ﬁnancial support from the National Marine Fisheries Service, NOAA. The ﬁndings and
conclusions of the NOAA author(s) in the paper are those of the NOAA author(s) and do not
necessarily represent the views of the reviewers nor the National Marine Fisheries Service, NOAA.
Any use of trade, product, or ﬁrm names does not imply an endorsement by the U.S. Government.
References
Anselin, L. (1988), Spatial Econometrics: Methods and Models, Dordrecht, the Netherlands: Kluwer
Academic Publishers.
Anselin, L., Syabri, I., and Kho, Y. (2006), “GeoDa: an introduction to spatial data analysis,”
Geographical analysis, 38, 5–22.
13

Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2004), Hierarchical Modeling and Analysis for
Spatial Data, Boca Raton, FL, USA: Chapman and Hall/CRC.
Besag, J. (1974), “Spatial Interaction and the Statistical Analysis of Lattice Systems (with discus-
sion),” Journal of the Royal Statistical Society, Series B, 36, 192–236.
— (1986), “On the statistical analysis of dirty pictures,” Journal of the Royal Statistical Society.
Series B (Methodological), 259–302.
Besag, J. and Higdon, D. (1999), “Bayesian analysis of agricultural ﬁeld experiments,” Journal of
the Royal Statistical Society: Series B (Statistical Methodology), 61, 691–746.
Besag, J. and Kooperberg, C. (1995), “On conditional and intrinsic autoregressions,” Biometrika,
82, 733–746.
Bivand, R., Hauke, J., and Kossowski, T. (2013), “Computing the Jacobian in Gaussian spatial
autoregressive models: An illustrated comparison of available methods,” Geographical Analysis,
45, 150–179.
Bivand, R. and Piras, G. (2015), “Comparing Implementations of Estimation Methods for Spatial
Econometrics,” Journal of Statistical Software, 63, 1–36.
Brook, D. (1964), “On the distinction between the conditional probability and the joint probability
approaches in the speciﬁcation of nearest-neighbour systems,” Biometrika, 51, 481–483.
Clayton, D. and Kaldor, J. (1987), “Empirical Bayes estimates of age-standardized relative risks
for use in disease mapping,” Biometrics, 43, 671–681.
Cliﬀord, P. (1990), “Markov random ﬁelds in statistics,” in Disorder in Physical Systems: A Volume
in Honour of John M. Hammersley, eds. Grimmett, R. G. and Welsh, D. J. A., New York, NY,
USA: Oxford University Press, pp. 19–32.
Cressie, N. A. C. (1993), Statistics for Spatial Data, Revised Edition, New York: John Wiley &
Sons.
Cullis, B. and Gleeson, A. (1991), “Spatial analysis of ﬁeld experiments-an extension to two di-
mensions,” Biometrics, 1449–1460.
Dormann, C. F., McPherson, J. M., Ara´ujo, M. B., Bivand, R., Bolliger, J., Carl, G., Davies,
R. G., Hirzel, A., Jetz, W., Kissling, W. D., K¨uhn, I., Ohlem¨uller, R., Peres-Neto, P. R., Reinek-
ing, B., Schr¨oder, B., Schurr, F. M., and Wilson, R. (2007), “Methods to account for spatial
autocorrelation in the analysis of species distributional data: a review,” Ecography, 30, 609–628.
Golub, G. H. and Van Loan, C. F. (2013), Matrix Computations, Fourth Edition, Baltimore: John
Hopkins University Press.
Haining, R. (1990), Spatial Data Analysis in the Social and Environmental Sciences, Cambridge,
UK: Cambridge University Press.
Hammersley, J. M. and Cliﬀord, P. (1971), “Markov ﬁelds on ﬁnite graphs and lattices,” Unpub-
lished Manuscript.
14

Harville, D. A. (1997), Matrix Algebra from a Statistician’s Perspective, New York, NY: Springer.
Hoyer, P. O. (2004), “Non-negative matrix factorization with sparseness constraints,” Journal of
Machine Learning Research, 5, 1457–1469.
Kent, J. T. (2011), “Discussion on the paper by Lindgren, Rue and Lindstr¨om,” Journal of the
Royal Statistical Society: Series B (Statistical Methodology), 73, 423–498.
Kirkpatrick, S., Gelatt, C. D., Vecchi, M. P., et al. (1983), “Optimization by simulated annealing,”
Science, 220, 671–680.
Kissling, W. D. and Carl, G. (2008), “Spatial autocorrelation and the selection of simultaneous
autoregressive models,” Global Ecology and Biogeography, 17, 59–71.
Lawson, A. B. (2013), Statistical Methods in Spatial Epidemiology, Chichester, UK: John Wiley &
Sons.
LeSage, J. and Pace, R. K. (2009), Introduction to Spatial Econometrics, Boca Raton, FL, USA:
Chapman and Hall/CRC.
Li, H., Calder, C. A., and Cressie, N. (2007), “Beyond Moran’s I: testing for spatial dependence
based on the spatial autoregressive model,” Geographical Analysis, 39, 357–375.
Li, S. Z. (2009), Markov random ﬁeld modeling in image analysis, Springer Science & Business
Media.
Lichstein, J. W., Simons, T. R., Shriner, S. A., and Franzreb, K. E. (2002), “Spatial autocorrelation
and autoregressive models in ecology,” Ecological Monographs, 72, 445–463.
Lindgren, F., Rue, H., and Lindstr¨om, J. (2011), “An explicit link between Gaussian ﬁelds and
Gaussian Markov random ﬁelds: the stochastic partial diﬀerential equation approach,” Journal
of the Royal Statistical Society: Series B (Statistical Methodology), 73, 423–498.
Mardia, K. V. (1990), “Maximum likelihood estimation for spatial models,” in Spatial Statistics:
Past, Present and Future, ed. Griﬃth, D., Michigan Document Services, Ann Arbor, MI, USA:
Institute of Mathematical Geography, Monograph Series, Monograph #12, pp. 203–253.
Martin, R. (1987), “Some comments on correction techniques for boundary eﬀects and missing
value techniques,” Geographical Analysis, 19, 273–282.
Patterson, H. and Thompson, R. (1974), “Maximum likelihood estimation of components of vari-
ance,” in Proceedings of the 8th International Biometric Conference, Biometric Society, Wash-
ington, DC, pp. 197–207.
Patterson, H. D. and Thompson, R. (1971), “Recovery of inter-block information when block sizes
are unequal,” Biometrika, 58, 545–554.
R Core Team (2016), R: A Language and Environment for Statistical Computing, R Foundation
for Statistical Computing, Vienna, Austria.
15

Rue, H. and Held, L. (2005), Gauss Markov Random Fields: Theory and Applications, Boca Raton,
FL, USA: Chapman and Hall/CRC.
Rue, H., Martino, S., and Chopin, N. (2009), “Approximate Bayesian inference for latent Gaussian
models by using integrated nested Laplace approximations,” Journal of the Royal Statistical
Society: Series B (Statistical Methodology), 71, 319–392.
Schabenberger, O. and Gotway, C. A. (2005), Statistical Methods for Spatial Data Analysis, Boca
Raton, Florida: Chapman Hall/CRC.
Wall, M. M. (2004), “A close look at the spatial structure implied by the CAR and SAR models,”
Journal of Statistical Planning and Inference, 121, 311–324.
Waller, L. A. and Gotway, C. A. (2004), Applied Spatial Statistics for Public Health Data, John
Wiley and Sons, New Jersey.
Whittle, P. (1954), “On stationary processes in the plane,” Biometrika, 41, 434–449.
16

APPENDIX: Propositions on Weights Matrices
The following proposition is used to show condition C1 for CAR models.
Proposition 3. Let Σ = AM, where Σ, A, and M are square matrices, Σ is symmetric, A−1
exists, and M is symmetric and positive deﬁnite. Then Σ is positive deﬁnite if and only if all of
the eigenvalues of A are positive real numbers.
Proof. (⇐=): Let M−1/2 be the matrix such that M−1/2M−1/2 = M−1, and let M1/2 be the matrix
such that M1/2M1/2 = M. Now, A = ΣM−1 = M1/2[M−1/2ΣM−1/2]M−1/2. Then A has the
same eigenvalues as [M−1/2ΣM−1/2] because they are similar matrices (Harville, 1997, p. 525). If
Σ is positive deﬁnite, then [M−1/2ΣM−1/2] is positive deﬁnite, so all eigenvalues of A are positive
real numbers.
(=⇒): Let A = UΛU−1, where the columns of U contain orthonormal eigenvectors and Λ
is a diagonal matrix of eigenvalues that are all positive and real. Because of symmetry, AM =
MAT = M(UT )−1ΛUT , so AM(UT )−1 = M(UT )−1Λ. This shows that both U and M(UT )−1
have columns that contain the eigenvectors for A, so each column in U has a corresponding column
in M(UT )−1 that is a scalar multiple. Let Γ be a diagonal matrix of those scalar multiples, so that
M(UT )−1 = UΓ. Hence, U−1M(UT )−1 = Γ, and notice that all diagonal elements of Γ will be
positive because M is positive deﬁnite. Also U−1M = ΓUT , so Σ = AM = UΛU−1M = UΛΓUT .
Because Λ and Γ are diagonal, each with all positive real values, Σ is positive deﬁnite.
Condition C1 is satisﬁed by letting ΣCAR in (4) be Σ in Proposition 3, by letting (I −C)−1
in (3) be A in Proposition 3 (note that if (I −C)−1 has all positive eigenvalues, so too does I −C),
and by letting M in (4) be M in Proposition 3.
Next, we show the conditions on ρ that ensure that (I−ρW) has either nonzero eigenvalues,
or positive eigenvalues.
Proposition 4. Consider the square matrix (I −ρW), where wi,i = 0. Let {λi} be the set of
eigenvalues of W, and suppose all eigenvalues are real. Then
(i) if ρ /∈{λ−1
i } for all nonzero λi, then (I −ρW) is nonsingular, and
(ii) assume at least two eigenvalues of W are not zero, and let λ[1] be the smallest eigenvalue of
W, and λ[N] be the largest eigenvalue of W. If 1/λ[1] < ρ < 1/λ[N], then (I −ρW) has only
positive eigenvalues.
Proof. Let ω be an eigenvalue of (I −ρW), so the following holds,
(I −ρW)x = ωx.
(A.1)
Let vi be the eigenvector corresponding to λi. Solving for ωi in (A.1), let x = vi. Then,
vi −ρWvi = ωivi,
=⇒vi −ρλivi = ωivi,
=⇒(1 −ρλi)vi = ωivi,
=⇒(1 −ρλi) = ωi.
Then,
17

(i) (I −ρW) is nonsingular if all ωi ̸= 0, so if λi = 0, then ωi = 1 for all ρ, otherwise (1 −ρλi) ̸=
0 =⇒ρ ̸= 1/λi for all nonzero λi.
(ii) For all ωi > 0, (1 −ρλi) > 0 =⇒1 > ρλi for all i. If λi < 0, then 1/λi < ρ, and if λi > 0,
then ρ < 1/λi. For all negative λi, only 1/λ[1] < ρ will ensure all (1 −ρλi) > 0, and for
positive λi, only ρ < 1/λ[N] will ensure all (1 −ρλi) > 0. Hence 1/λ[1] < ρ < 1/λ[N] will
guarantee that all eigenvalues of (I −ρW) are positive.
18
