Nilpotent & Idempotent Matrices: The Weirdest Matrices #tgt

Nilpotent & Idempotent Matrices Nilpotent Matrices Definition: A square matrix A is said to be nilpotent if A^n = 0 for some positive integer n. Properties: - Nilpotent matrices have all eigenvalues equal to 0. - The determinant of a nilpotent matrix is 0. - Nilpotent matrices are not invertible. Idempotent Matrices Definition: A square matrix A is said to be idempotent if A^2 = A$. Properties: - Idempotent matrices have eigenvalues equal to 0 or 1. - The determinant of an idempotent matrix is either 0 or 1. - Idempotent matrices can be used in projection matrices. *Key Differences* - Nilpotent matrices have all eigenvalues equal to 0, while idempotent matrices have eigenvalues equal to 0 or 1. - Nilpotent matrices are not invertible, while idempotent matrices can be invertible if they are full rank. #matrix #tgt #tgtmath