When is a symmetric matrix positive definite

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Ask Question. Asked 6 years, 6 months ago. Active 10 months ago. Viewed 1k times. Are symmetric matrices always at least positive semi-definite? Thanks so much, Jhance. Add a comment. Active Oldest Votes. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. Skip to content A real matrix is symmetric positive definite if it is symmetric is equal to its transpose, and By making particular choices of in this definition we can derive the inequalities Satisfying these inequalities is not sufficient for positive definiteness.

For example, the matrix satisfies all the inequalities but for. Two equivalent conditions to being symmetric positive definite are every leading principal minor , where the submatrix comprises the intersection of rows and columns to , is positive, the eigenvalues of are all positive. Here are some other important properties of symmetric positive definite matrices.

Examples of symmetric positive definite matrices, of which we display only the instances, are the Hilbert matrix the Pascal matrix and minus the second difference matrix, which is the tridiagonal matrix All three of these matrices have the property that is non-decreasing along the diagonals. A symmetric positive definite matrix that was often used as a test matrix in the early days of digital computing is the Wilson matrix What is the best way to test numerically whether a symmetric matrix is positive definite?

Symmetric block matrices often appear in applications. If is nonsingular then we can write which shows that is congruent to a block diagonal matrix, which is positive definite when its diagonal blocks are.

Consider, as an example, the matrix which has leading principal minors , , and and a negative eigenvalue. References This is a minimal set of references, which contain further useful references within. Nicholas J. Higham, Computing a nearest symmetric positive semidefinite matrix , Linear Algebra Appl.