Sufficiency of Signed Principal Minors for Semidefiniteness:
A Relatively Easy Proof
David M. Mandy
Department of Economics
University of Missouri
118 Professional Building
Columbia, MO 65203 USA
[email protected]
573-882-1763
September 23, 2016
Abstract. Extant proofs that signed principal minors is sufficient for semidefiniteness of a Hermitian matrix are relatively inaccessible to economics audiences. The paper presents a new proof
that relies only on facts familiar to economics graduate students.
Keywords. Semidefinite, Hermitian, Matrix, Principal Minors.
JEL Code. C02.
1
1
Introduction
Characterizing semidefiniteness of a matrix is a frequent need in econometrics and economic theory. Sometimes the matrix is definite and this is established with relative ease by studying the
signs of leading principal minors. However some contexts require that the semidefinite case be
considered explicitly. For example, duality for price-taking economic actors requires that profit
(cost/expenditure) be a linear homogeneous and convex (concave) function of prices. Linear homogeneity implies the Hessian is singular (when it exists), whence the matrix under study in
these settings cannot be definite. Another example is necessary second order conditions for an
extremum. The necessary, but not sufficient, conditions are implied by maximizing behavior so
semidefiniteness is all that can be inferred from the postulated behavior.
Although it is well-known that all principal minors must be signed to establish semidefiniteness, proofs in the economics literature are scarce. Standard mathematical economics books including Chiang [2, pp. 320 – 323], de la Fuente [4, p. 270], Lancaster [8, pp. 297 – 300], Simon and
Blume [9, Theorem 16.2], Sydsæter et al. [10, Theorem 1.7.1] and Takayama [11, Theorem 1.E.11]
do not prove the semidefinite case. Extant proofs typically work directly with the quadratic form
or characteristic equation and use a limit argument [3] [1, p. 223] [5, p. 307], making them relatively inaccessible to many economists.
This short note provides an alternate proof that signing all of the principal minors is sufficient
for semidefiniteness of a Hermitian matrix. The method of proof enables relatively easy consideration of many situations with a singular matrix. The proof relies on six facts about matrices which,
for clarity of the background needed to understand the proof, are listed at the outset. These six
facts have long been familiar to economics graduate students. For example, they are all succinctly
proven in Chapter 4 of Johnston [7].1 With these six facts in hand, the proof herein uses nothing
more than simple observations about the entries in the matrix.
2
Notation and Reminders
s
• x = y means the real numbers x and y have the same sign.
• ᾱ denotes the complex conjugate of α ∈ C1 . ᾱα is real and nonnegative, and is zero if and
only if α = 0.
• Given a matrix A with complex entries aij , A∗ denotes the conjugate transpose of A (the (j, i)
entry of A∗ is āij ). If A ∈ Cn (a vector) then A∗ A is real and nonnegative, and is zero if and
only if A = 0.
• Given an n × n complex matrix A, let J ⊂ {1, . . . , n} denote an index set of the rows and
columns of A (J 6= ∅). AJ denotes the principal submatrix of A consisting of the rows
1
Johnston works only with real matrices. Everything herein is stated for complex matrices because doing so adds
little complication.
2
and columns in J, and #J denotes the number of elements in J (the order of the principal
submatrix). For the special cases J = {1, . . . , i} for i = 1, . . . , n, AJ is denoted Ai (i.e., Ai is
the i × i leading principal submatrix of A, and An = A). A0 ≡ 1 for notational convenience.
The determinant |AJ | is a principal minor of order #J. Enumerating the elements of J in
ascending order as J = {j1 , . . . , j#J } yields a permutation matrix PJ with entries pij = 0
for i, j = 1, . . . , n except piji = 1 for i = 1, . . . , #J and pii = 1 for i = #J + 1, . . . , n.
When symmetrically applied to A, PJ makes AJ a leading principal submatrix. As with all
permutation matrices, PJ∗ PJ = In .
• A is Hermitian (symmetric, in the real case) if A∗ = A. A Hermitian matrix is square and has
real diagonal entries.
• An n × n matrix A is positive (negative) definite if x∗ Ax > (<) 0 for every nonzero vector
x ∈ Cn . If the defining inequality is weak then A is positive (negative) semidefinite. Implicit in
this definition is that the product x∗ Ax is real for every nonzero x ∈ Cn .2
3
Six Familiar Facts about Complex Matrices
1. (AB)∗ = B ∗ A∗ for conformable matrices A and B.
2. |αA| = αn |A| for α ∈ C1 and an n × n matrix A.
3. If a square matrix A is partitioned as:
A11 A12
A=
,
A21 A22
where A11 is square (i.e., a leading principal submatrix) and nonsingular, then:
|A| = |A11 ||A22 − A21 A−1
11 A12 |.
4. Every positive (semi) definite (Hermitian) matrix has a unique positive (semi) definite (Hermitian) square root matrix.
5. (The definite case; necessity) Every principal minor of a positive definite (Hermitian) matrix
is positive.
6. (The definite case; sufficiency) If every leading principal minor of a Hermitian matrix is
positive then the matrix is positive definite.
A is Hermitian if and only if x∗ Ax is real for every x ∈ Cn [6, Theorem 4.1.4]. We do not use this fact directly but it
shows that semidefinite matrices are always Hermitian.
2
3
4
Main Theorem
Theorem 1. If the n × n Hermitian matrix A has a principal submatrix AJ satisfying:
1. AJ is positive (negative) definite,
2. No higher order principal submatrix of A is positive (negative) definite, and
3. |AK | ≥ 0 (has sign (−1)#K or is zero) for every K satisfying J ⊂ K ⊂ {1, . . . , n};
then A is positive (negative) semidefinite.
Proof. Begin with the positive semidefinite case. If AJ = A then A is trivially positive semidefinite,
so assume AJ excludes some rows and (the same) columns. Then:
AJ B
∗
PJ APJ =
B∗ C
for some matrices B and C (C is Hermitian). Consider an arbitrary column bi from B and the
corresponding diagonal entry cii from C. Using |AJ | > 0 from item 1 (Fact 5) and Fact 3 yields:
AJ bi s
∗ −1
∗ −1
b∗ cii = |AJ ||cii − bi AJ bi | = cii − bi AJ bi .
i
This is nonnegative due to item 3. If it is positive then the principal submatrix under consideration
is positive definite (Facts 5 and 6), contradicting item 2, so cii − b∗i A−1
J bi = 0. Now consider any
two columns from B and the corresponding 2 × 2 submatrix from C:
AJ bi bj ∗
∗
cii cij
b
−1
i
b
i cii cij = |AJ | c∗ cjj − b∗ AJ bi bj j
ij
b∗ c∗ cjj j
ij
bj 0
cij − b∗i A−1
s J
= ∗
0
cij − b∗j A−1
J bi
∗
∗ −1
= −(cij − b∗i A−1
J bj )(cij − bj AJ bi )
∗ −1
∗
= −(cij − b∗i A−1
J bj )(cij − bi AJ bj ) .
The last equality uses Fact 1. This is again nonnegative due to item 3, which implies cij −b∗i A−1
J bj =
−1
∗
0h for every i, ji(including i = j). Therefore C = B AJ B. Now use Fact 4 to define F =
1/2
AJ
−1/2
AJ
B and using Fact 1 note that:
"
#
1/2
h
i A
B
AJ
AJ B
J
1/2
−1/2
F F =
=
=
.
AJ
AJ B
−1/2
B∗ C
B ∗ B ∗ A−1
B ∗ AJ
J B
∗
4
For any nonzero x ∈ Cn , let y(x) = F PJ x. Then, again using Fact 1:
x∗ Ax = x∗ PJ∗ F ∗ F PJ x = y(x)∗ y(x) ≥ 0.
Hence A is positive semidefinite.
For the negative semidefinite case, replace AJ in the above argument by −AJ and A by −A,
noting that (1) −AJ is positive definite, (2) no higher order principal submatrix of −A is positive
definite, and (3) | − AK | = (−1)#K |AK | ≥ 0 for every K satisfying J ⊂ K ⊂ {1, . . . , n} (Fact 2).
The conclusion is −A is positive semidefinite, or A is negative semidefinite.
5
Applications
Theorem 1 readily gives the sufficient condition for semidefiniteness.
Corollary 1. If every principal minor of a Hermitian matrix A is nonnegative (of order #J has sign
(−1)#J or is zero) then A is positive (negative) semidefinite.
Proof. If every diagonal entry of A is zero then:
aii aij = −aij āij for every i, j = 1, . . . , n.
0≤
āij ajj This implies aij = 0 for every i, j. That is, A is a matrix of zeros, and is therefore trivially positive
(negative) semidefinite. So assume there is a nonzero diagonal entry. That entry is a positive (negative) definite principal submatrix, so A has a highest order positive (negative) definite principal
submatrix AJ satisfying the assumptions of Theorem 1.
In most duality applications even though the matrix under consideration is singular there is an
(n−1) dimensional submatrix that is definite. That definiteness is verified by checking only (n−1)
leading principal minors, and is therefore much easier than attempting to verify semidefiniteness
directly with Corollary 1. Theorem 1 shows that doing so suffices to establish semidefiniteness of
the original matrix.
Corollary 2. Assume A is an n × n singular Hermitian matrix. If A has an (n − 1)st order positive
(negative) definite principal submatrix AJ then A is positive (negative) semidefinite.
Proof. The only principal submatrix of higher order than AJ is A, and |A| = 0. Apply Theorem
1.
Acknowledgements. I am grateful to Jonathan Hamilton and J. Isaac Miller for helpful discussions.
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References
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[4] de la Fuente, A. Mathematical Methods and Models for Economists. Cambridge University
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[5] Gantmacher, FR. The Theory of Matrices, Vol. 1. Chelsea, New York (1959).
[6] Horn, RA and CR Johnson. Matrix Analysis (second ed.). Cambridge University Press, Cambridge (2013).
[7] Johnston, J. Econometric Methods (third ed.). McGraw-Hill, New York (1984).
[8] Lancaster, K. Mathematical Economics. Macmillian, London (1970).
[9] Simon, CP and L. Blume. Mathematics for Economists. Norton, New York (1994).
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[11] Takayama, A. Mathematical Economics (second ed.). Cambridge University Press, Cambridge (1985).
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