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1 Definition
2 Examples
3 Characterizations
- 3.1 Quadratic forms
4 Negative-definite, semidefinite and indefinite matrices
5 Further properties
6 Non-Hermitian matrices
7 See also
8 Notes
9 References

Definition

An n × n real symmetric matrix M is positive definite if z^TMz > 0 for all non-zero vectors z with real entries ( $z \in \mathbb{R}^n$ ), where z^T denotes the transpose of z.

For complex matrices, this definition becomes: a Hermitian matrix M is positive definite if z^†Mz > 0 for all non-zero complex vectors z, where z^† denotes the conjugate transpose of z. The quantity z^†Mz is always real because M is a Hermitian matrix. For this reason, positive-definite matrices are often defined to be Hermitian matrices satisfying z^†Mz > 0 for non-zero z. The section Non-Hermitian matrices discusses the consequences of dropping the requirement that M be Hermitian.

Examples

The matrix $M_0 = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$ is positive definite. For a vector with entries $\textbf{z}= \begin{bmatrix} z_0 \\ z_1\end{bmatrix}$ the quadratic form is $\begin{bmatrix} z_0 & z_1\end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} z_0 \\ z_1\end{bmatrix}=z_0^2+z_1^2;$ when the entries z₀, z₁ are real and at least one of them nonzero, this is positive.

The matrix $M_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$ is not positive definite. When $\textbf{z}= \begin{bmatrix} 1\\ -1\end{bmatrix}$ the quadratic form at z is then

$\begin{bmatrix} 1 & -1\end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ -1\end{bmatrix}=-2 < 0.$

Another example of positive definite matrix is given by

$A = \begin{bmatrix} 2&-1&0\\-1&2&-1\\0&-1&2 \end{bmatrix}.$

It is positive definite since for any non-zero vector $x = \begin{bmatrix} x_1\\x_2\\x_3 \end{bmatrix}$ , we have

$x^{t}Ax = \begin{bmatrix} x_1&x_2&x_3 \end{bmatrix} \begin{bmatrix} 2&-1&0\\-1&2&-1\\0&-1&2 \end{bmatrix} \begin{bmatrix} x_1\\x_2\\x_3 \end{bmatrix}$ $= 2x_1^{2} - 2x_1x_2 + 2x_2^{2} - 2x_2x_3 + 2x_3^{2}$ $= x_1^{2}+(x_1 - x_2)^{2} + (x_2 - x_3)^{2}+x_3^{2}> 0.$

For a huge class of examples, consider that in Statistics positive definite matrices appears as covariance matrices. In fact all positive definite matrices are a covariance matrix for some probability distribution.

Characterizations

Let M be an n × n Hermitian matrix. The following properties are equivalent to M being positive definite:

1. All eigenvalues

λ i

M

are positive. Recall that any Hermitian M, by the spectral theorem, may be regarded as a real diagonal matrix D that has been re-expressed in some new coordinate system (i.e.,

M = P - 1 D P

for some unitary matrix P whose rows are orthonormal eigenvectors of M, forming a basis). So this characterization means that M is positive definite if and only if the diagonal elements of D (the eigenvalues) are all positive. In other words, in the basis consisting of the eigenvectors of M, the action of M is component-wise multiplication with a (fixed) element in Cⁿ with positive entries^{[clarification needed]}.
2. The sesquilinear form $\langle \mathbf{x},\mathbf{y}\rangle := \mathbf{y}^*\mathbf{M}\mathbf{x}$

defines an inner product on Cⁿ. (In fact, every inner product on Cⁿ arises in this fashion from a Hermitian positive definite matrix.)

3. M is the Gram matrix of some collection of linearly independent vectors $\textbf{x}_1,\ldots,\textbf{x}_n \in \mathbb{C}^k$

for some k. That is, M satisfies:

$M_{ij} = \langle \textbf{x}_i, \textbf{x}_j\rangle = \textbf{x}_i^{*} \textbf{x}_j.$

The vectors x_i may optionally be restricted to fall in Cⁿ. In other words, M is of the form A*A where A is not necessarily square but must be injective in general.

4. All the following matrices have a positive determinant (Sylvester's criterion):

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 $M$
...
$M$ itself

In other words, all of the leading principal minors are positive. For positive semidefinite matrices, all principal minors have to be non-negative. The leading principal minors alone do not imply positive semidefiniteness, as can be seen from the example

$\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \end{bmatrix}.$
5. There exists a unique lower triangular matrix

L

, with strictly positive diagonal elements, that allows the factorization of

M

into

M = L L *

where $L *$ is the conjugate transpose of $L$ . This factorization is called Cholesky decomposition.

For real symmetric matrices, these properties can be simplified by replacing $\mathbb{C}^n$ with $\mathbb{R}^n$ , and "conjugate transpose" with "transpose."

Quadratic forms

Echoing condition 2 above, one can also formulate positive-definiteness in terms of quadratic forms. Let K be the field R or C, and V be a vector space over K. A Hermitian form

$B : V \times V \rightarrow K$

is a bilinear map such that B(x, y) is always the complex conjugate of B(y, x). Such a function B is called positive definite if B(x, x) > 0 for every nonzero x in V.

Negative-definite, semidefinite and indefinite matrices

The n × n Hermitian matrix M is said to be negative-definite if

$x^{*} M x < 0\,$

for all non-zero $x \in \mathbb{C}^n$ (or, all non-zero $x \in \mathbb{R}^n$ for the real matrix).

It is called positive-semidefinite (or sometimes nonnegative-definite) if

$x^{*} M x \geq 0$

for all $x \in \mathbb{C}^n$ (or, all $x \in \mathbb{R}^n$ for the real matrix).

It is called negative-semidefinite if

$x^{*} M x \leq 0$

for all $x \in \mathbb{C}^n$ (or, all $x \in \mathbb{R}^n$ for the real matrix).

A matrix M is positive-semidefinite if and only if it arises as the Gram matrix of some set of vectors. In contrast to the positive-definite case, these vectors need not be linearly independent.

For any matrix A, the matrix A^*A is positive semidefinite, and rank(A) = rank(A^*A). Conversely, any positive semidefinite matrix M can be written as M = A^*A; this is the Cholesky decomposition.

A Hermitian matrix which is neither positive- nor negative-semidefinite is called indefinite.

A matrix is negative definite if all kth order leading principal minors are negative if k is odd and positive if k is even.

A matrix is negative-definite, negative-semidefinite, or positive-semidefinite if and only if all of its eigenvalues are negative, non-positive, or non-negative, respectively.

Further properties

If $M$ is positive-semidefinite, one sometimes writes $M \geq 0$ and if $M$ is positive-definite one writes $M > 0$ .[1]The notion comes from functional analysis where positive definite matrices define positive operators.

For arbitrary square matrices $M, N$ we write $M\geq N$ if $M-N \geq 0$ , i.e. $M - N$ is positive semi-definite. This defines a partial ordering on the set of all square matrices. One can similarly define a strict partial ordering $M > N$ .

Every positive definite matrix is invertible and its inverse is also positive definite.[2] If $M \geq N > 0$ then $N^{-1} \geq M^{-1} > 0$ .[3]

2. If

M

is positive definite and

r > 0

is a real number, then

r M

is positive definite.[4]

If $M$ and $N$ are positive definite, then the sum $M + N$ [4] and the products $M N M$ and $N M N$ are also positive definite. If $M N = N M$ , then $M N$ is also positive definite.

3. If

M = (m i j) > 0

then the diagonal entries

m i i

are real and positive. As a consequence tr

(M) > 0

. Furthermore $| m_{ij} | \leq \sqrt{m_{ii} m_{jj}} \leq \frac{m_{ii}+m_{jj}}{2}$ and thus $\max |m_{ij}| \leq \max|m_{ii}|$

[5]

4. A matrix

M

is positive semi-definite if and only if there is a positive semi-definite matrix

B

with

B 2 = M

. This matrix B is unique[6], is called the square root of M, and is denoted with

B = M 1 / 2

(the square root B is not to be confused with the matrix L in the Cholesky factorization M = LL*, which is also sometimes called the square root of M). If

M > N > 0

then

M 1 / 2 > N 1 / 2 > 0

.
5. If

M, N > 0

then $M\otimes N > 0.$ (Here $\otimes$ denotes Kronecker product.)
6. For matrices

M = (m i j)

and

N = (n i j)

, write M ˆ N for the entry-wise product of

M

and

N

, i.e. the matrix whose

i, j

entry is

m i j n i j

. Then M ˆ N is the Hadamard product of

M

and

N

. The Hadamard product of two positive-definite matrices is again positive-definite and the Hadamard product of two positive-semidefinite matrices is again positive-semidefinite (this result is often called the Schur product theorem).[7] Furthermore, if M and N are positive-semidefinite, then the following inequality, due to Oppenheim, holds: $\det(M\circ N) \geq (\det N) \prod_{i} m_{ii}.$ [8]
7. Let