Positive-definite matrix

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In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form (or a sesquilinear form in the complex case).

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Equivalent formulations

Let M be an n × n Hermitian matrix. In the following we denote the transpose of a matrix or vector <math>a</math> by <math>a^{T}</math>, and the conjugate transpose by <math>a^{*}</math>. The matrix M is said to be positive definite if it has one (and therefore all) of the following 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>.

Note that the quantity <math>z^{*} M z</math> is always real.

2. All eigenvalues <math>\lambda_i</math> of <math>M</math> are positive. (Recall that the eigenvalues of a Hermitian matrix are necessarily real).
3. 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.)

4. All the following matrices (the leading principal minors) have a positive determinant (the Sylvester criterion):
  • 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

Analogous statements hold if M is a real symmetric matrix, by replacing <math>\mathbb{C}^n</math> by <math>\mathbb{R}^n</math>, and the conjugate transpose by the transpose.

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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 the sum <math>M + N</math> and the products <math>MNM</math> and <math>NMN</math> are 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.

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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.

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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.

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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>B : 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>.

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References

it:Matrice definita positiva he:מטריצה חיובית fi:Positiivisesti definiitti matriisi

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