For a matrix \(A\), and its eigenvalue \(\lambda_i\)
 \(A\) and \(A^T\) always share the same eigenvalues
 \(A\) and \(M^{1}AM\) always share the same eigenvalues
 eigenvalues of \(A^{1}\) are \(1/\lambda_i\)
 eigenvalues of \(A+cI\) are \(\lambda_i + c\)
 eigenvectors of \(AA^T\) are basis of columns of \(A\)
 eigenvectors of \(A^TA\) are basis of rows of \(A\)
 if \(A\) is singular, then some \(\lambda_i = 0\)
 if \(A\) is symmetric,
 \(\lambda_i\) are all real
 eigenvectors can be chosen orthonormal
 number of \(\lambda_i > 0\) = number of positive pivots
 if \(A\) is skewsymmetric, then \(\lambda_i\) are all imaginary
 if \(A\) is symmetric positive definite, then all \(\lambda_i > 0\)
 if \(A\) is full rank, then eigenvectors form a basis for \(\mathbb{R}^n\)
 if \(A\) is real, then eigenvalues and eigenvectors come in conjugate pairs
 if \(A\) is diagonal, then eigenvalues are the diagonal elements

if \(A\) is orthogonal, then $$ \lambda_i =1\(for all\)i$$  if \(A\) is a projector matrix (i.e. \(A^2=A\)), then \(\lambda_i\) is either 0 or 1
 if \(A\) is a Markov/stochastic matrix, \(\max_i \lambda_i = 1\) (which the corresponding eigenvector is the vector of stationary probabilities)