The statement is imprecise: eigenvectors corresponding to distinct eigenvalues of a symmetric matrix must be orthogonal to each other. Eigenvectors corresponding to the same eigenvalue …

This topic explains how to use the chol and eig functions to determine whether a matrix is symmetric positive definite (a symmetric matrix with all positive eigenvalues).

Jun 28, 2016 · So, the question is how many entries can be chosen indepently in a symmetric matrix of order n? 2) How many entries can be chosen indepently in a skew-symmetric matrix

The skew-symmetric matrices have arbitrary elements on one side with respect to the diagonal, and those elements determine the other triangle of the matrix. So they are in number of $ (n^2 …

25 All the proofs here use algebraic manipulations. But I think it may be more illuminating to think of a symmetric matrix as representing an operator consisting of a rotation, an anisotropic …

Eigenvalues play an important role in situations where the matrix is a trans-formation from one vector space onto itself. Systems of linear ordinary differential equations are the primary …

Oct 25, 2020 · A real symmetric matrix is always orthogonally diagonalizable, meaning that there's a basis for $\mathbb R^n$ consisting of mutually perpendicular eigenvectors of the matrix. …

Oct 13, 2017 · Actually it's clearly harder to diagonalize a 3x3 symmetric matrix (giving its 3 real eigenvalues) than it is to find the determinant (product of the eigenvalues).

A square matrix, A, is skew-symmetric if it is equal to the negation of its nonconjugate transpose, A = -A.'. In terms of the matrix elements, this means that

Please explain All positive integral powers of a Symmetric Matrix gives a Symmetric Matrix. Odd positive integral powers of a Skew-Symmetric Matrix give a Skew-Symmetric Matrix.