Sylvester's criterion

In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester.

Sylvester's criterion states that a Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant:

• the upper left 1-by-1 corner of M,
• the upper left 2-by-2 corner of M,
• the upper left 3-by-3 corner of M,
• ${}\quad \vdots$
• M itself.

In other words, all of the leading principal minors must be positive.

An analogous theorem holds for characterizing positive-semidefinite Hermitian matrices, except that it is no longer sufficient to consider only the leading principal minors: a Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative.^[1]

Proof

The proof is only for nonsingular Hermitian matrix with coefficients in $\mathbb {R}$, therefore only for nonsingular real-symmetric matrices.

Positive definite or semidefinite matrix: A symmetric matrix A whose eigenvalues are positive (λ > 0) is called positive definite, and when the eigenvalues are just nonnegative (λ ≥ 0), A is said to be positive semidefinite.

Theorem I: A real-symmetric matrix A has nonnegative eigenvalues if and only if A can be factored as A = B^TB, and all eigenvalues are positive if and only if B is nonsingular.^[2]

Proof:

Forward implication: If A ∈ R^n×n is symmetric, then, by the spectral theorem, there is an orthogonal matrix P such that A = PDP^T , where D = diag (λ1, λ2, . . . , λn) is real diagonal matrix with entries being eigenvalues of A and P is such that its columns are the eigenvectors of A. If λi ≥ 0 for each i, then D^1/2 exists, so A = PDP^T = PD^1/2D^1/2P^T = B^TB for B = D^1/2P^T, and λi > 0 for each i if and only if B is nonsingular. Reverse implication: Conversely, if A can be factored as A = B^TB, then all eigenvalues of A are nonnegative because for any eigenpair (λ, x): $\lambda ={{\frac {x^{T}Ax}{x^{T}x}}}={{\frac {x^{T}B^{T}Bx}{x^{T}x}}}={{\frac {\|Bx\|^{2}}{\|x\|^{2}}}}\geq 0.$

Theorem II (The Cholesky decomposition): The symmetric matrix A possesses positive pivots if and only if A can be uniquely factored as A = R^TR, where R is an upper-triangular matrix with positive diagonal entries. This is known as the Cholesky decomposition of A, and R is called the Cholesky factor of A.^[3]

Proof:

Forward implication: If A possesses positive pivots (therefore A possesses an LU factorization: A = L·U'), then, it has an LDU factorization A = LDU = LDL^T in which D = diag(u11, u22, . . . , unn) is the diagonal matrix containing the pivots uii > 0. ${\begin{aligned}{\mathbf {A}}&=LU'={\begin{bmatrix}1&0&\cdots &0\\\ell _{{12}}&1&\cdots &0\\\vdots &\vdots &&\vdots \\\ell _{{1n}}&\ell _{{2n}}&\cdots &1\end{bmatrix}}{\begin{bmatrix}u_{{11}}&u_{{12}}&\cdots &u_{{1n}}\\0&u_{{22}}&\cdots &u_{{2n}}\\\vdots &\vdots &&\vdots \\0&0&\cdots &u_{{nn}}\end{bmatrix}}\\[8pt]&=LDU={\begin{bmatrix}1&0&\cdots &0\\\ell _{{12}}&1&\cdots &0\\\vdots &\vdots &&\vdots \\\ell _{{1n}}&\ell _{{2n}}&\cdots &1\end{bmatrix}}{\begin{bmatrix}u_{{11}}&0&\cdots &0\\0&u_{{22}}&\cdots &0\\\vdots &\vdots &&\vdots \\0&0&\cdots &u_{{nn}}\end{bmatrix}}{\begin{bmatrix}1&u_{{12}}/u_{{11}}&\cdots &u_{{1n}}/u_{{11}}\\0&1&\cdots &u_{{2n}}/u_{{22}}\\\vdots &\vdots &&\vdots \\0&0&\cdots &1\end{bmatrix}}\end{aligned}}$ By a uniqueness property of the LDU decomposition, the symmetry of A yields: U = L^T, consequently A = LDU = LDL^T. Setting R = D^1/2L^T where D^1/2 = diag($\scriptstyle {\sqrt {u_{{11}}}},\scriptstyle {\sqrt {u_{{22}}}},\ldots ,\scriptstyle {\sqrt {u_{{11}}}}$) yields the desired factorization, because A = LD^1/2D^1/2L^T = R^TR, and R is upper triangular with positive diagonal entries. Reverse implication: Conversely, if A = RR^T, where R is lower triangular with a positive diagonal, then factoring the diagonal entries out of R is as follows: ${\mathbf {R}}=LD={\begin{bmatrix}1&0&\cdots &0\\r_{{12}}/r_{{11}}&1&\cdots &0\\\vdots &\vdots &&\vdots \\r_{{1n}}/r_{{11}}&r_{{2n}}/r_{{22}}&\cdots &1\end{bmatrix}}{\begin{bmatrix}r_{{11}}&0&\cdots &0\\0&r_{{22}}&\cdots &0\\\vdots &\vdots &&\vdots \\0&0&\cdots &r_{{nn}}\end{bmatrix}}.$ R = LD, where L is a lower triangular matrix with a unit diagonal and D is the diagonal matrix whose diagonal entries are the rii ’s. Hence D has a positive diagonal and hence D is non-singular. Hence D^2 is a non-singular diagonal matrix. Also, L^T is an upper triangular matrix with a unit diagonal. Consequently, A = LD^2L^T is the LDU factorization for A, and thus the pivots must be positive because they are the diagonal entries in D^2. Uniqueness of the Cholesky decomposition: If we have another Cholesky decomposition A = R1R1^T of A, where R1 is lower triangular with a positive diagonal, then similar to the above we may write R1 = L1D1, where L1 is a lower triangular matrix with a unit diagonal and D1 is a diagonal matrix whose diagonal entries are the same as the corresponding diagonal entries of R1. Consequently, A = L1D1^2L1^T is an LDU factorization for A. By the uniqueness of the LDU factorization of A, we have L1 = L and D1^2 = D^2. As both D1 and D are diagonal matrices with positive diagonal entries, we have D1 = D. Hence R1 = L1D1 = LD = R. Hence A has a unique Cholesky decomposition.

Theorem III: Let A_k be the k × k leading principal submatrix of A_n×n. If A has an LU factorization A = LU, where L is a lower triangular matrix with a unit diagonal, then det(A_k) = u₁₁u₂₂ · · · u_kk, and the k-th pivot is u_kk = det(A₁) = a₁₁ for k = 1, u_kk = det(A_k)/det(A_k−1) for k = 2, 3, . . . , n, where u_jj is the (j, j)-th entry of U for all j = 1, 2, . . . , n.^[4]

Combining Theorem II with Theorem III yields:

Statement I: If the symmetric matrix A can be factored as A=R^TR where R is an upper-triangular matrix with positive diagonal entries, then all the pivots of A are positive (by Theorem II), therefore all the leading principal minors of A are positive (by Theorem III).

Statement II: If the nonsingular n × n symmetric matrix A can be factored as $A=B^{T}B$, then the QR decomposition (closely related to Gram-Schmidt process) of B (B = QR) yields: $A=B^{T}B=R^{T}Q^{T}QR=R^{T}R$, where Q is orthogonal matrix and R is upper triangular matrix.

As A is non-singular and $A=R^TR$, it follows that all the diagonal entries of R are non-zero. Let r_jj be the (j, j)-th entry of E for all j = 1, 2, . . . , n. Then r_jj ≠ 0 for all j = 1, 2, . . . , n.

Let F be a diagonal matrix, and let f_jj be the (j, j)-th entry of F for all j = 1, 2, . . . , n. For all j = 1, 2, . . . , n, we set f_jj = 1 if r_jj > 0, and we set f_jj = -1 if r_jj < 0. Then $F^TF=I_n$, the n × n identity matrix.

Let S=FR. Then S is an upper-triangular matrix with all diagonal entries being positive. Hence we have $A=R^TR=R^TF^TFR=S^TS$, for some upper-triangular matrix S with all diagonal entries being positive.

Namely Statement II requires the non-singularity of the symmetric matrix A.

Combining Theorem I with Statement I and Statement II yields:

Statement III: If the real-symmetric matrix A is positive definite then A possess factorization of the form A = B^TB, where B is nonsingular (Theorem I), the expression A = B^TB implies that A possess factorization of the form A = R^TR where R is an upper-triangular matrix with positive diagonal entries (Statement II), therefore all the leading principal minors of A are positive (Statement I).

In other words, Statement III proves the "only if" part of Sylvester's Criterion for non-singular real-symmetric matrices.

Sylvester's Criterion: The real-symmetric matrix A is positive definite if and only if all the leading principal minors of A are positive.

Notes

1. Carl D. Meyer, Matrix Analysis and Applied Linear Algebra. See chapter 7.6 Positive Definite Matrices, page 566
2. Carl D. Meyer, Matrix Analysis and Applied Linear Algebra. See chapter 7.6 Positive Definite Matrices, page 558
3. Carl D. Meyer, Matrix Analysis and Applied Linear Algebra. See chapter 3.10 The LU Factorization, Example 3.10.7, page 154
4. Carl D. Meyer, Matrix Analysis and Applied Linear Algebra. See chapter 6.1 Determinants, Exercise 6.1.16, page 474

References

• Gilbert, George T. (1991), "Positive definite matrices and Sylvester's criterion", The American Mathematical Monthly, Mathematical Association of America, 98 (1): 44–46, doi:10.2307/2324036, ISSN 0002-9890, JSTOR 2324036.
• Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6. See Theorem 7.2.5.
• Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 0-89871-454-0.