Скачать или смотреть The Matrices Which are Usually Unusual - Idempotent, Involutory, Nilpotent Matrix

What is Metrix?
Matrix is an arrangement of numbers into rows and columns. ... A matrix is a rectangular arrangement of numbers into rows and columns. For example, matrix A has two rows and three columns.

Idempotent Matrix: An idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix A is idempotent if and only if A^{2}=A. For this product A^{2} to be defined, A must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings.

Involutory Matrix: An involutory matrix is a matrix that is its own inverse. That is, multiplication by matrix A is an involution if and only if A^2 = I. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any non-singular matrix multiplied by its inverse is the identity.

Nilpotent Matrix: A nilpotent matrix is a square matrix N such that
N^{k}=0,
for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N.

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