A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of … See more

Let A be a square n × n matrix with n linearly independent eigenvectors qi (where i = 1, ..., n). Then A can be factored as where Q is the square n × n matrix whose ith column is the eigenvector qi of A, and Λ is the … See more

The eigendecomposition allows for much easier computation of power series of matrices. If f (x) is given by then we know that Because Λ is a diagonal matrix, functions of Λ are very easy to calculate:
The off-dia… See more

Spectral matrices are matrices that possess distinct eigenvalues and a complete set of eigenvectors. This characteristic allows spectral matrices to be fully diagonalizable, meaning they can be decomposed into simpler for… See more

• The product of the eigenvalues is equal to the determinant of A Note that each eigenvalue is raised to the power ni, the algebraic multiplicity.
• The sum of the eigenvalues is equal to the trace of A Note that each ei… See more

Suppose that we want to compute the eigenvalues of a given matrix. If the matrix is small, we can compute them symbolically using the characteristic polynomial. However, this is often impossible for larger matrices, i… See more