Properties of Eigenvalue

Consider a square matrix A with eigenvalues λ₁, λ₂ … λ_n

• The determination of A is a product of all its eigenvalue. [det(A) = λ₁ × λ₂….λ_n]
• Matrix A is invertible if and only if every eigenvalue is non-zero.
• Eigenvalue of real symmetric and Hermitian matrices are equal.
• Eigenvalue of real skew-symmetric and skew Hermitian matrices are either pure or zero.
• Eigenvalue of unitary and orthogonal matrices are of unit modulus |λ| = 1.
• Eigen value of A^-1= 1/λ₁,1/λ₂,… 1/λ_n.
• Eigen value of A^k = λ^k₁, λ^k₂, …. λ^k_n
• If A and B are two matrices of the same order then the eigenvalue of AB = Eigenvalue of BA.
• If a square matrix A is a lower/upper triangular matrix, then its eigenvalue will be the diagonal elements of the matrix.