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# positive definite

# Introduction

The definiteness of a matrix is an important property that has use in many areas of mathematics and even physics. Below are some examples:

1. In optimizing problems, the definiteness of the Hessian matrix determines the quality of an extremal value. The full details can be found on this page.

Definition [1] Suppose $A$ is an $n\times n$ square Hermitian matrix. If, for any non-zero vector $x$, we have that

$x^{\ast}Ax>0,$ |

then $A$ a *positive definite* matrix. (Here $x^{\ast}=\overline{x}^{t}$,
where $\overline{x}$ is the complex conjugate of $x$, and $x^{t}$ is
the transpose of $x$.)

One can show that a Hermitian matrix is positive definite if and only if all its eigenvalues are positive [1]. Thus the determinant of a positive definite matrix is positive, and a positive definite matrix is always invertible. The Cholesky decomposition provides an economical method for solving linear equations involving a positive definite matrix. Further conditions and properties for positive definite matrices are given in [2].

# References

- 1
M. C. Pease,
*Methods of Matrix Algebra*, Academic Press, 1965 - 2
C.R. Johnson,
*Positive definite matrices*, American Mathematical Monthly, Vol. 77, Issue 3 (March 1970) 259-264.

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15A48*no label found*

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## Comments

## positive definite matrices

What is the quickest way (visually) to determine whether a matrix is positive definite or not? Can it be done by inspection? (i.e., for matrices with both positive and negative values)? Is a matrix with a positive determinant necessarily positive definite?

## Re: positive definite matrices

In general, I don't believe there is any quick and easy way to

determine whether a matrix is positive definite. At least

without any prior knowledge of the matrix.

MathWorld gives some necessary conditions on the elements of a

matrix for the matrix to be positive definite. These are, however,

not sufficient. Therefore, using these one can - at most - prove that

a matrix is not positive definite. See

http://mathworld.wolfram.com/PositiveDefiniteMatrix.html

One of the conditions is that the determinant should be positive.

This is quite easy to understand: A matrix is positive definite

if and only if all it's eigenvalues are positive. Since the determinant

is the product of the eigenvalues the result follows.

Probably the easiest way to determine if a matrix is positive definite

is the calculate the determinants of all upper-left (or lower-right) sub

matrices. If all determinants are positive, (and only then) the matrix

is positive definite (again, see MathWorld). For example,

the two-two matrix

a b

c d

is positive definite if and only if ad-cb > 0 and a > 0.

Hope this helps.

Matte

## Re: positive definite matrices

Well, each matrix it have associated an caracteristic equation. I think that through its invariants coefficients we can know if that matrix if positive definite or not. What do you think?

## Re: positive definite matrices

Can you provide an example? I am aware that matrices have characteristic equations, but it is not clear what you mean by "through its invariants coefficients we can know if the matrix is positive definite or not"...

## Re: positive definite matrices

Well I said "I think...", but is not true. The right way was given by Mr. matte. Like you know the characteristic equation is the

Cayley-Hamilton or secular equation. By definition, a matrix is said to be positive if it has positive principal values and linearly independent principal vectors. Positive matrices are said to be positive definite if they are also coefficients (the cause of my confusion!)of positive definite quadratic forms. In particular, a positive symmetric matrix is also positive definite.

Pedro

## Re: positive definite matrices

Thank you for the clarification. My interest is towards understanding covariance matrices a little better. I read somewhere that examples of positive definite matrices are covariance matrices, but I felt that this was wrong since a covariance matrix is in no way constrained to have positive diagonal elements (one of the 'necessary but not sufficient conditions').

To your knowledge are there graphical methods of assessing the stability of a covariance matrix obtained from data for representing a particular distribution? Say you have data which is approximately multivariate gaussian, and you form the covariance matrix; then can one use a spectral portrait analysis to assess how 'stable' this inference is -- that is, is there some useful way of interpreting 'perturbations of a covariance matrix estimated from data' with respect to the expected distribution?

## Re: positive definite matrices

Hi Mr. silence,

Unfortunately I am not specialist in statistcs nor in data analysis techniques. Surely you know the results of Silverstein [1]. I don't know if there is somebody in PlanetMath that can help it. Nevertheless, I have seen in the Web an important amount of sites related to the subject of its interest exists.

Regards,

Pedro

[1] J.W. Silverstein. Eigenvalues and eigenvectors of large dimensional sample covariance matrices. Contemporary Mathematics,

50:153-159,1986.