Orthogonally Diagonalizable
Symmetric Matrices and Eigenvalues
How to Orthogonally Diagonalize
Quadratic Forms
Definiteness
Principal Axes
Ellipses and HyperbolasHow to Examples Exercise
How to:
Definiteness is terminology used to classify symmetric matrices. Then are several different ways of determining the definiteness of A and we explain them all here.
(1) A symmetric matrix A is positive definite if:
(i) q = dot(x, A*x) > 0 for all non-zero x
(ii) All of A's eigenvalues are strictly positive
(iii) det(Am) > 0 for all m = 1, ..., n. Here Am is defined as the MxM matrix obtained by omitting all rows and columns past the m-th.
Similarly we can define A to be positive semidefinite, negative definite, or negative semidefinite. The qualficiations for these differ only slightly from positive definite:
(2) Positive semidefinite - change "> 0" and "positive" to ">= 0" and "non-negative" in the definition above
(3) Negative definite - change "> 0" and "positive" to "< 0" and "negative" in the definition above
(4) Negative semidefinite - change "> 0" and "positive" to "<= 0" and "non-positive" in the definition above
There is one final type of definiteness, indefinite.
(5) A symmetric matrix A is indefinite if:
(i) q = dot(x, A*x) takes positive AND negative values
(ii) A has positive AND negative eigenvalues
(iii) Am takes positive AND negative determinants for some m.
Examples:
Exercise: