# Positive-definite matrix

In linear algebra, the positive-definite matrices are (in several ways) analogous to the positive real numbers. First, define some things:

• [itex]a^{T}[itex] is the transpose of a matrix or vector [itex]a[itex]
• [itex]a^{*}[itex] is the complex conjugate of its transpose [itex]a[itex]
• [itex]\mathbb{R}[itex] is the set of all real numbers
• [itex]\mathbb{C}[itex] is the set of all complex numbers
• [itex]\mathbb{Z}[itex] is the set of all integers
• [itex]M[itex] is any Hermitian matrix

An n × n Hermitian matrix [itex]M[itex] is said to be positive definite if it has one (and therefore all) of the following six equivalent properties:

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

Every positive definite matrix is invertible and its inverse is also positive definite. If [itex]M[itex] is positive definite and [itex]r > 0[itex] is a real number, then [itex]r M[itex] is positive definite. If [itex]M[itex] and [itex]N[itex] are positive definite, then [itex]M + N[itex] is also positive definite, and if [itex]M N = N M[itex], then [itex]MN[itex] is also positive definite. Every positive definite matrix [itex]M[itex], has at least one square root matrix [itex]N[itex] such that [itex]N^2 = M[itex]. In fact, [itex]M[itex] may have infinitely many square roots, but exactly one positive definite square root.

## Negative-definite, semidefinite and indefinite matrices

The Hermitian matrix [itex]M[itex] is said to be negative-definite if

[itex]x^{*} M x < 0[itex]

for all non-zero [itex]x \in \mathbb{R}^n[itex] (or, equivalently, all non-zero [itex]x \in \mathbb{C}^n[itex]). It is called positive-semidefinite if

[itex]x^{*} M x \geq 0[itex]

for all [itex]x \in \mathbb{R}^n[itex] (or [itex]\mathbb{C}^n[itex]) and negative-semidefinite if

[itex]x^{*} M x \leq 0[itex]

for all [itex]x \in \mathbb{R}^n[itex] (or [itex]\mathbb{C}^n[itex]).

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

## Non-Hermitian matrices

A real matrix M may have the property that xTMx > 0 for all nonzero real vectors x without being symmetric. The matrix

[itex] \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} [itex]

provides an example. In general, we have xTMx > 0 for all real nonzero vectors x if and only if the symmetric part, (M + MT) / 2, is positive definite.

The situation for complex matrices may be different, depending on how one generalizes the inequality z*Mz > 0. If z*Mz is real for all complex vectors z, then the matrix M is necessarily Hermitian. So, if we require that z*Mz be real and positive, then M is automatically Hermitian. On the other hand, we have that Re(z*Mz) > 0 for all complex nonzero vectors z if and only if the Hermitian part, (M + M*) / 2, is positive definite.

There is no agreement in the literature on the proper definition of positive-definite for non-Hermitian matrices.

## Generalizations

Suppose [itex]K[itex] denotes the field [itex]\mathbb{R}[itex] or [itex]\mathbb{C}[itex], [itex]V[itex] is a vector space over [itex]K[itex], and [itex] : V \times V \rightarrow K[itex] is a bilinear map which is Hermitian in the sense that [itex]B(x, y)[itex] is always the complex conjugate of [itex]B(y, x)[itex]. Then [itex]B[itex] is called positive definite if [itex]B(x, x) > 0[itex] for every nonzero [itex]x[itex] in [itex]V[itex].

## References

• Art and Cultures
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