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Symmetric matrices, quadratic forms, matrix norm, and SVD • eigenvectors of symmetric matrices • quadratic forms • inequalities for quadratic forms • positive semidefinite matrices • norm of a matrix • singular value decomposition 15–1
Symmetric matrices appear in geometry, for example, when introducing more general dot products v Av or in statistics as correlation matrices Cov[X k,X l] or in quantum mechanics as observables or in neural networks as learning maps x sign(Wx) or in graph theory as adjacency matrices.
EXHIBITION. "Where do symmetric matrices occur?" Some informal motivation: I) PHYSICS: In quantum mechanics a system is described with a vector v(t) which depends on time t. The evolution is given by the Schroedinger equation v_ = i hLv, where L is a symmetric matrix and h is a small number called the Planck constant.
1 Symmetric Matrices We review some basic results concerning symmetric matrices. All matrices that we discuss are over the real numbers. Let Abe a real, symmetric matrix of size d dand let Idenote the d didentity matrix. Perhaps the most important and useful property of symmetric matrices is that their eigenvalues behave very nicely.
1. Real Symmetric Matrices The most common matrices we meet in applications are symmetric, that is, they are square matrices which are equal to their transposes. In symbols, At = A. Examples. • 12 22 ‚; 2 4 1¡10 ¡102 023 3 5 are symmetric, but 2 4 122 013 004 3 5; 2 4 01¡1 ¡102 1¡20 3 5 are not. Symmetric matrices are in many ways much ...
A symmetric matrix is a matrix A such that AT = A. Such a matrix is necessarily square. Its main diagonal entries are arbitrary, but its other entries occur in pairs, on opposite sides of the main diagonal. Example 1. Determine which matrix is symmetric.
This lecture covers section 6.4 of the textbook. Today we’re going to look at diagonalizing a matrix when the matrix is symmetric. It turns out that symmetric matrices have a number of totally awesome properties: The eigenvalues of a symmetric matrix are all real.
