Linear Algebra | Definitions of Periodic idempotent and Nilpotent Matrices | GCUF | PU | BSCS | BSMT

periodic matrix:
A square matrix such that the matrix power for a positive integer is called a periodic matrix. If is the least such integer, then the matrix is said to have period.

idempotent matrix:
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix is idempotent if and only if . For this product to be defined, must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings.

nilpotent matrix:
A square matrix A is called a nilpotent matrix, if Am=0 for some positive integer m.

nilpotent index definition:
A square matrix A is called nilpotent if there is a non-negative integer k such that Ak is the zero matrix.
The smallest such an integer k is called degree or index of A.