In mathematics,Sylvester’s criterion is anecessary and sufficient criterion to determine whether aHermitian matrix ispositive-definite, due toJames Joseph Sylvester.

Sylvester's criterion states that an ×n Hermitian matrixM is positive-definiteif and only if all the following matrices have a positivedeterminant:

• the upper left 1-by-1 corner ofM,
• the upper left 2-by-2 corner ofM,
• the upper left 3-by-3 corner ofM,
• ${\displaystyle ⋮}$
• M itself.

In other words, all of theleadingprincipal minors must be positive. By using appropriate permutations of rows and columns ofM, it can also be shown that the positivity ofany nested sequence ofn principal minors ofM is equivalent toM being positive-definite.^[1]

An analogous theorem holds for characterizingpositive-semidefinite Hermitian matrices, except that it is no longer sufficient to consider only theleading principal minors as illustrated by the Hermitian matrix

A Hermitian matrixM is positive-semidefinite if and only ifallprincipal minors ofM are nonnegative.^[2][3]

Suppose${\displaystyle M_n}$ is${\displaystyle n\times n}$ Hermitian matrix${\displaystyle M_n^{†}=M_n}$. Let${\displaystyle M_k,k=1,\dots n}$ be the leading principal minor matrices, i.e. the${\displaystyle k\times k}$ upper left corner matrices. It will be shown that if${\displaystyle M_n}$ is positive definite, then the principal minors are positive; that is,${\displaystyle det{M}_k>0}$ for all${\displaystyle k}$.

${\displaystyle M_k}$ is positive definite. Indeed, choosing

${\displaystyle x=\left(\begin{array}{c}{x}_1\\ ⋮\\ x_k\\ 0\\ ⋮\\ 0\end{array}\right)=\left(\begin{array}{c}\overrightarrow{x}\\ 0\\ ⋮\\ 0\end{array}\right)}$

we can notice that Equivalently, the eigenvalues of${\displaystyle M_k}$ are positive, and this implies that${\displaystyle det{M}_k>0}$ since the determinant is the product of the eigenvalues.

To prove the reverse implication, we useinduction. The general form of an${\displaystyle (n+1)\times (n+1)}$ Hermitian matrix is

,

where${\displaystyle M_n}$ is an${\displaystyle n\times n}$ Hermitian matrix,${\displaystyle \overrightarrow{v}}$ is a vector and${\displaystyle d}$ is a real constant.

Suppose the criterion holds for${\displaystyle M_n}$. Assuming that all the principal minors of${\displaystyle M_{n+1}}$ are positive implies that${\displaystyle det{M}_{n+1}>0}$,${\displaystyle det{M}_n>0}$, and that${\displaystyle M_n}$ is positive definite by the inductive hypothesis. Denote

${\displaystyle x=\left(\begin{array}{c}\overrightarrow{x}\\ x_{n+1}\end{array}\right)}$

then

${\displaystyle x^{†}{M}_{n+1}x={\overrightarrow{x}}^{†}{M}_n\overrightarrow{x}+x_{n+1}{\overrightarrow{x}}^{†}\overrightarrow{v}+{\overline{x}}_{n+1}{\overrightarrow{v}}^{†}\overrightarrow{x}+d|x_{n+1}{|}^2}$

By completing the squares, this last expression is equal to

${\displaystyle ({\overrightarrow{x}}^{†}+{\overrightarrow{v}}^{†}{M}_n^{-1}{\overline{x}}_{n+1})M_n(\overrightarrow{x}+x_{n+1}{M}_n^{-1}\overrightarrow{v})-|x_{n+1}{|}^2{\overrightarrow{v}}^{†}{M}_n^{-1}\overrightarrow{v}+d|x_{n+1}{|}^2}$

${\displaystyle =(\overrightarrow{x}+\overrightarrow{c}{)}^{†}{M}_n(\overrightarrow{x}+\overrightarrow{c})+|x_{n+1}{|}^2(d-{\overrightarrow{v}}^{†}{M}_n^{-1}\overrightarrow{v})}$

where${\displaystyle \overrightarrow{c}=x_{n+1}{M}_n^{-1}\overrightarrow{v}}$ (note that${\displaystyle M_n^{-1}}$ exists because the eigenvalues of${\displaystyle M_n}$ are all positive.) The first term is positive by the inductive hypothesis. We now examine the sign of the second term. By using theblock matrix determinant formula

on${\displaystyle (\ast )}$ we obtain

${\displaystyle det{M}_{n+1}=det{M}_n(d-{\overrightarrow{v}}^{†}{M}_n^{-1}\overrightarrow{v})>0}$, which implies${\displaystyle d-{\overrightarrow{v}}^{†}{M}_n^{-1}\overrightarrow{v}>0}$.

Consequently,${\displaystyle x^{†}{M}_{n+1}x>0.}$

Let${\displaystyle M_n}$ be ann xn Hermitian matrix. Suppose${\displaystyle M_n}$ is semidefinite. Essentially the same proof as for the case that${\displaystyle M_n}$ is strictly positive definite shows that all principal minors (not necessarily the leading principal minors) are non-negative.

For the reverse implication, it suffices to show that if${\displaystyle M_n}$ has all non-negative principal minors, then for allt>0, all leading principal minors of the Hermitian matrix${\displaystyle M_n+t{I}_n}$ are strictly positive, where${\displaystyle I_n}$ is thenxnidentity matrix. Indeed, from the positive definite case, we would know that the matrices${\displaystyle M_n+t{I}_n}$ are strictly positive definite. Since the limit of positive definite matrices is always positive semidefinite, we can take${\displaystyle t\to 0}$ to conclude.

To show this, let${\displaystyle M_k}$ be thekth leading principal submatrix of${\displaystyle M_n.}$ We know that${\displaystyle q_k(t)=det(M_k+t{I}_k)}$ is a polynomial int, related to the characteristic polynomial${\displaystyle p_{M_k}(t)}$ via$${\displaystyle q_k(t)=(-1{)}^k{p}_{M_k}(-t).}$$ We use the identity inCharacteristic polynomial#Properties to write$${\displaystyle q_k(t)=\sum _{j=0}^k{t}^{k-j}\mathrm{tr}\left(\stackrel{j}{\bigwedge }{M}_k\right),}$$where$\mathrm{tr}\left(\stackrel{j}{\bigwedge }{M}_k\right)$ is the trace of thejth exterior power of${\displaystyle M_k.}$

Fromthe definition of minors, we know that the entries in the matrix expansion of${\displaystyle \stackrel{j}{\bigwedge }{M}_k}$ (forj > 0) are just the minors of${\displaystyle M_k.}$ In particular, the diagonal entries are the principal minors of${\displaystyle M_k}$, which of course are also principal minors of${\displaystyle M_n}$, and are thus non-negative. Since the trace of a matrix is the sum of the diagonal entries, it follows that$${\displaystyle \mathrm{tr}\left(\stackrel{j}{\bigwedge }{M}_k\right)\ge 0.}$$ Thus the coefficient of${\displaystyle t^{k-j}}$ in${\displaystyle q_k(t)}$ is non-negative for allj > 0. Forj = 0, it is clear that the coefficient is 1. In particular,${\displaystyle q_k(t)>0}$ for allt > 0, which is what was required to show.

• Gilbert, George T. (1991), "Positive definite matrices and Sylvester's criterion",The American Mathematical Monthly,98 (1), Mathematical Association of America: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. Theorem 7.2.5.
• Carl D. Meyer (June 2000),Matrix Analysis and Applied Linear Algebra,SIAM,ISBN 0-89871-454-0.

• Matrix theory

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