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 skew-symmetric, 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)