positive-definite matrix

In linear algebra, the positive-definite matrices are (in several ways) analogous to the positive real numbers. An n × n Hermitian matrix

M

is said to be positive definite if it has one (and therefore all) of the following six equivalent properties. First, define some things:

align="top"\| 1.	For all non-zero vectors $z \in \mathbb{C}^n$ we have $\textbf{z}^{} M \textbf{z} > 0$ . Here we view $z$ as a column vector with $n$ complex entries and $z^{}$ as the complex conjugate of its transpose. ( $z^{*} M z$ is always real.)
align="top"\| 2.	For all non-zero vectors $x$ in $\mathbb{R}^n$ we have $\textbf{x}^{T} M \textbf{x} > 0$
align="top"\| 3.	For all non-zero vectors $u \in \mathbb{Z}^n$ , we have $\textbf{u}^{T} M \textbf{u} > 0$ .
align="top"\| 4.	All eigenvalues of $M$ are positive. $\lambda_i(M) > 0 \; \forall i$
align="top"\| 5.	The form $\langle \textbf{x},\textbf{y}\rangle = \textbf{x}^{*} M \textbf{y}$ defines an inner product on $\mathbb{C}^n$ . (In fact, every inner product on $\mathbb{C}^n$ arises in this fashion from a Hermitian positive definite matrix.)
align="top"\| 6.	All the following matrices have 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 $M$ ... $M$ itself

Further properties

Every positive definite matrix is invertible and its inverse is also positive definite. If

M

is positive definite and

r > 0

is a real number, then

r M

is positive definite. If

M

and

N

are positive definite, then

M + N

is also positive definite, and if

M N = N M

, then

MN

is also positive definite. Every positive definite matrix

M

, has at least one square root matrix

N

such that

N^2 = M

. In fact,

M

may have infinitely many square roots, but exactly one positive definite square root.

The Hermitian matrix

M

is said to be negative-definite if

x^{*} M x < 0

for all non-zero

x \in \mathbb{R}^n

(or, equivalently, all non-zero

x \in \mathbb{C}^n

). It is called positive-semidefinite if

x^{*} M x \geq 0

for all

x \in \mathbb{R}^n

(or

\mathbb{C}^n

) and negative-semidefinite if

x^{*} M x \leq 0

for all

x \in \mathbb{R}^n

(or

\mathbb{C}^n

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

Suppose

K

denotes the field

\mathbb{R}

\mathbb{C}

V

K

, and

: V \times V \rightarrow K

is a bilinear map which is Hermitian in the sense that

B(x, y)

is always the complex conjugate of

B(y, x)

. Then

B

is called positive definite if

B(x, x) > 0

for every nonzero

x

V

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