Idempotent Matrix Formula
Let us consider a “2 × 2” square matrix . As P is an idempotent matrix, P2 = P.
Now, comparing the terms on each side, we get
1) a2 + bc = a
bc = a − a2
2) ab + bd = b
ab + bd − b = 0
b (a + d − 1) = 0
b = 0 or a + d − 1 = 0
d = 1 − a
So, if a matrix is said to be an idempotent matrix, if bc = a − a2 and d = 1 − a.
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.