spectral decomposition

(1.7 hours to learn)


The spectral decomposition is the decomposition of a symmetric matrix A into QDQ^T, where Q is an orthogonal matrix and D is a diagonal matrix. The columns of Q correspond to the eigenvectors of A, and the diagonal entries of D correspond to the eigenvalues. This is possible because of a surprising fact about symmetric matrices: they have a full set of orthogonal eigenvectors. This decomposition gives a useful way to think about symmetric matrices: they are like diagonal matrices in a rotated coordinate system.


This concept has the prerequisites:

Core resources (read/watch one of the following)


MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
Author: Gilbert Strang


See also