Properties of Idempotent Matrix
The following are some important properties of an idempotent matrix:
- Every idempotent matrix is a square matrix.
- All idempotent matrices are singular matrices, apart from the identity matrix.
- The determinant of an idempotent matrix is either one or zero.
- The non-diagonal entries of an idempotent matrix can be non-zero entries.
- The trace of an idempotent matrix is always an integer and equal to the rank of the matrix.
- The eigenvalues of an idempotent matrix are either zero or one.
- The following is the relationship between idempotent and involuntary matrices: A square matrix “A” is said to be an idempotent matrix if and only if P = 2A − I is an involuntary matrix.
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Idempotent Matrix
A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an “m by n” matrix when it has “m” rows and “n” columns and is written as an “m × n” matrix. For example, a matrix of order “5 × 6” has five rows and six columns. We have different types of matrices like rectangular matrices, square matrices, null matrices, triangular matrices, symmetric matrices, etc.