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)