Eigenvectors

For a square matrix A of size n

• The product of the eigenvalues is equal to the determinant of A.

• The sum of the eigenvalues is equal to the trace of A.

• Singular matrix have zero eigenvalues.

• maximum number of eigenvalues is n.

• Eigenvectors with distinct Eigenvalues are linearly independent.

• rank is equal to the number of non-zero eigen values

• Eigen decomposition $X=UDU^{-1}$

Further, for a symmetric matrix of size n

• Eigenvectors of real symmetric matrices ( Hermitian ) for distinct eigenvalues are orthogonal.

• A real symmetric matrices (Hermitian matrix) is negative-definite, negative-semidefinite, or positive-semidefinite if and only if all of its eigenvalues are negative, non-positive, or non-negative, respectively.

• Eigen decomposition $X=UDU^{T}$, where columns of $U$ are orthogonal vectors.

Eigen Decomposition Theorem