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Symmetric matrices play the same role as the real numbers do among the complex numbers. Their eigenvalues often have physical or geometrical interpretations. One can also calculate with symmetric matrices like with numbers: for example, we can solve B2 = A for B if A is symmetric matrix and B is square root of A.) This is not possible in ...
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
if A is a symmetric matrix with real entries, then the roots of its charac-teristic equation are all real. Example 1. The characteristic equations of • 01 10 ‚ and • 0 ¡1 10 ‚ are ‚2 ¡1 = 0 and ‚2 +1=0 respectively. Notice the dramatic efiect of a simple change of sign. The reason for the reality of the roots (for a real ...
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.
Session Overview. Special matrices have special eigenvalues and eigenvectors. Symmetric and positive definite matrices have extremely nice properties, and studying these matrices brings together everything we’ve learned about pivots, determinants and eigenvalues.
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.
Symmetric Matrices and Quadratic Forms. Quadratic form. Suppose is a column vector in R , and is a symmetric × matrix. The term is called a quadratic form. The result of the quadratic form is a scalar. (1 × )( × )( × 1) The quadratic form is also called a quadratic function = The quadratic function’s input is the vector and the output is a. .
