In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix A {\displaystyle A} is idempotent if and only if A 2 = A {\displaystyle A^{2}=A} . For this product A 2 {\displaystyle A^{2}} to be defined, A {\displaystyle A} must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings.
Example Examples of 2 × 2 {\displaystyle 2\times 2} idempotent matrices are:
[ 1 0 0 1 ] [ 3 − 6 1 − 2 ] {\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\qquad {\begin{bmatrix}3&-6\\1&-2\end{bmatrix}}}
Examples of 3 × 3 {\displaystyle 3\times 3} idempotent matrices are:
[ 1 0 0 0 1 0 0 0 1 ] [ 2 − 2 − 4 − 1 3 4 1 − 2 − 3 ] {\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\qquad {\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}}}
Real 2 × 2 case If a matrix ( a b c d ) {\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}} is idempotent, then
a = a 2 + b c , {\displaystyle a=a^{2}+bc,}
b = a b + b d , {\displaystyle b=ab+bd,} implying b ( 1 − a − d ) = 0 {\displaystyle b(1-a-d)=0} so b = 0 {\displaystyle b=0} or d = 1 − a , {\displaystyle d=1-a,}
c = c a + c d , {\displaystyle c=ca+cd,} implying c ( 1 − a − d ) = 0 {\displaystyle c(1-a-d)=0} so c = 0 {\displaystyle c=0} or d = 1 − a , {\displaystyle d=1-a,}
d = b c + d 2 . {\displaystyle d=bc+d^{2}.}
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