FACTOID # 23: Wisconsin has more metal fabricators per capita than any other state.
 
 Home   Encyclopedia   Statistics   States A-Z   Flags   Maps   FAQ   About 
   
 
WHAT'S NEW
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


Encyclopedia > Positive definite matrix
Jump to: navigation, search

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). Jump to: navigation, search Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose - that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... Sesquilinear form - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

Contents


Equivalent formulations

Let M be an n × n Hermitian matrix. In the following we denote the transpose of a matrix of vector a by aT, and the conjugate transpose by a * . The matrix M is said to be positive definite if it has one (and therefore all) of the following equivalent properties: A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose - that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ... In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...

1. For all non-zero vectors z in mathbb{C}^n we have
textbf{z}^{*} M textbf{z} > 0.

Note that the quantity z * Mz is always real.

2. All eigenvalues λi of M are positive. (Recall that the eigenvalues of a Hermitian matrix are necessarily real).
3. The form
langle textbf{x},textbf{y}rangle = textbf{x}^{*} M textbf{y}

defines an inner product on mathbb{C}^n. (In fact, every inner product on mathbb{C}^n arises in this fashion from a Hermitian positive definite matrix.) In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... Jump to: navigation, search // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity α resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...

4. All the following matrices (the leading principle minors) have a positive determinant (the Sylvester criterion):
  • the upper left 1-by-1 corner of M
  • the upper left 2-by-2 corner of M
  • the upper left 3-by-3 corner of M
  • ...
  • M itself

Analogous statements hold if M is a real symmetric matrix, by replacing mathbb{C}^n by mathbb{R}^n, and the conjugate transpose by the transpose. Jump to: navigation, search In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... Jump to: navigation, search In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...


Further properties

Every positive definite matrix is invertible and its inverse is also positive definite. If M is positive definite and r > 0 is a real number, then rM is positive definite. If M and N are positive definite, then M + N is also positive definite, and if MN = NM, then MN is also positive definite. Every positive definite matrix M, has at least one square root matrix N such that N2 = M. In fact, M may have infinitely many square roots, but exactly one positive definite square root. Jump to: navigation, search In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is . ...


Negative-definite, semidefinite and indefinite matrices

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

x^{*} M x < 0,

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

x^{*} M x geq 0

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

x^{*} M x leq 0

for all x in mathbb{R}^n (or 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 xTMx > 0 for all nonzero real vectors x without being symmetric. The matrix

begin{bmatrix} 1 & 1  0 & 1 end{bmatrix}

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 K denotes the field mathbb{R} or mathbb{C}, V is a vector space over K, and B : V times V rightarrow K is a bilinear map which is Hermitian in the sense that B(x,y) is always the complex conjugate of B(y,x). Then B is called positive definite if B(x,x) > 0 for every nonzero x in V. Jump to: navigation, search In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...


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).

 
 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments

Want to know more?
Search encyclopedia, statistics and forums:

 


Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m