For a matrix , and its eigenvalue
 and always share the same eigenvalues
 and always share the same eigenvalues
 eigenvalues of are
 eigenvalues of are
 eigenvectors of are basis of columns of
 eigenvectors of are basis of rows of
 if is singular, then some
 if is symmetric,
 are all real
 eigenvectors can be chosen orthonormal
 number of = number of positive pivots
 if is skewsymmetric, then are all imaginary
 if is symmetric positive definite, then all
 if is full rank, then eigenvectors form a basis for
 if is real, then eigenvalues and eigenvectors come in conjugate pairs
 if is diagonal, then eigenvalues are the diagonal elements

if is orthogonal, then $$ \lambda_i =1i$$  if is a projector matrix (i.e. ), then is either 0 or 1
 if is a Markov/stochastic matrix, (which the corresponding eigenvector is the vector of stationary probabilities)