Positive-definite matrix

From Academic Kids

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

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

An n × n Hermitian matrix <math>M<math> 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 <math>z \in \mathbb{C}^n<math> we have
<math>\textbf{z}^{*} M \textbf{z} > 0<math>.

Here we view <math>z<math> as a column vector with <math>n<math> complex entries and <math>z^{*}<math> as the complex conjugate of its transpose. (<math>z^{*} M z<math> is always real.)

2. For all non-zero vectors <math>x<math> in

<math>\mathbb{R}^n<math> we have

<math>\textbf{x}^{T} M \textbf{x} > 0<math>
3. For all non-zero vectors <math>u \in \mathbb{Z}^n<math>, we have
<math>\textbf{u}^{T} M \textbf{u} > 0<math>.
4. All eigenvalues of <math>M<math> are positive.
<math>\lambda_i(M) > 0 \; \forall i<math>
5. The form
<math>\langle \textbf{x},\textbf{y}\rangle = \textbf{x}^{*} M \textbf{y}<math>

defines an inner product on <math>\mathbb{C}^n<math>. (In fact, every inner product on <math>\mathbb{C}^n<math> 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 <math>M<math>
  • the upper left 2-by-2 corner of <math>M<math>
  • the upper left 3-by-3 corner of <math>M<math>
  • ...
  • <math>M<math> itself
Contents

Further properties

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

Negative-definite, semidefinite and indefinite matrices

The Hermitian matrix <math>M<math> is said to be negative-definite if

<math>x^{*} M x < 0<math>

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

<math>x^{*} M x \geq 0<math>

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

<math>x^{*} M x \leq 0<math>

for all <math>x \in \mathbb{R}^n<math> (or <math>\mathbb{C}^n<math>).

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

<math> \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} <math>

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

References

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