Latent Semantic Analysis (Tutorial)

We will see that the number of eigenvalues is n for an n× n matrix. Regarding eigenvectors, if x is an eigenvector then so is ax for any scalar a. However, if we consider only one eigenvector for each ax family, then there is a 1-1 correspondence of such eigenvectors to eigenvalues. Typically, we consider eigenvectors of unit length. Diagonal matrices are simple, the eigenvalues are the entries on the diagonal, and the eigenvectors are their columns. For other matrices we find the eigenvalues first by reasoning as follows. If Ax = λx then (A−λI)x = 0, where I is the identity matrix. Since x is non-zero, matrix A−λI has dependent columns and thus its determinant |A − λI| must be zero. This gives us the equation |A − λI| = 0 whose solutions are the eigenvalues of A. As an example let