Property of Matrix and Its Eigenvalues

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 skew-symmetric, 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)