Definite matrix

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

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

An n × n Hermitian matrix ${\displaystyle M}$ 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 ${\displaystyle z\in \mathbb {C} ^{n}}$ we have ${\displaystyle {\textbf {z}}^{*}M{\textbf {z}}>0}$. Here we view ${\displaystyle z}$ as a column vector with ${\displaystyle n}$ complex entries and ${\displaystyle z^{*}}$ as the complex conjugate of its transpose. (${\displaystyle z^{*}Mz}$ is always real.)
2.	For all non-zero vectors ${\displaystyle x}$ in ${\displaystyle \mathbb {R} ^{n}}$ we have ${\displaystyle {\textbf {x}}^{T}M{\textbf {x}}>0}$
3.	For all non-zero vectors ${\displaystyle u\in \mathbb {Z} ^{n}}$, we have ${\displaystyle {\textbf {u}}^{T}M{\textbf {u}}>0}$.
4.	All eigenvalues of ${\displaystyle M}$ are positive. ${\displaystyle \lambda _{i}(M)>0\;\forall i}$
5.	The form ${\displaystyle \langle {\textbf {x}},{\textbf {y}}\rangle ={\textbf {x}}^{*}M{\textbf {y}}}$ defines an inner product on ${\displaystyle \mathbb {C} ^{n}}$. (In fact, every inner product on ${\displaystyle \mathbb {C} ^{n}}$ 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 ${\displaystyle M}$ the upper left 2-by-2 corner of ${\displaystyle M}$ the upper left 3-by-3 corner of ${\displaystyle M}$ ... ${\displaystyle M}$ itself

Further properties

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

Negative-definite, semidefinite and indefinite matrices

The Hermitian matrix ${\displaystyle M}$  is said to be negative-definite if

${\displaystyle x^{*}Mx<0}$ 

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

${\displaystyle x^{*}Mx\geq 0}$ 

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

${\displaystyle x^{*}Mx\leq 0}$ 

for all ${\displaystyle x\in \mathbb {R} ^{n}}$  (or ${\displaystyle \mathbb {C} ^{n}}$ ).

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 x^TMx > 0 for all nonzero real vectors x without being symmetric. The matrix

${\displaystyle {\begin{bmatrix}1&1\\0&1\end{bmatrix}}}$ 

provides an example. In general, we have x^TMx > 0 for all real nonzero vectors x if and only if the symmetric part, (M + M^T) / 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 ${\displaystyle K}$  denotes the field ${\displaystyle \mathbb {R} }$  or ${\displaystyle \mathbb {C} }$ , ${\displaystyle V}$  is a vector space over ${\displaystyle K}$ , and ${\displaystyle :V\times V\rightarrow K}$  is a bilinear map which is Hermitian in the sense that ${\displaystyle B(x,y)}$  is always the complex conjugate of ${\displaystyle B(y,x)}$ . Then ${\displaystyle B}$  is called positive definite if ${\displaystyle B(x,x)>0}$  for every nonzero ${\displaystyle x}$  in ${\displaystyle V}$ .

References

• Roger A. Horn and Charles R. Johnson. Matrix Analysis, Chapter 7. Cambridge University Press, 1985. ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).