In linear algebra, an idempotent matrix is a square matrix that, when multiplied by itself, yields itself as the result. In other words, it is a matrix that satisfies the equation A^2 = A.

One example of an idempotent matrix is the identity matrix, which is a square matrix with 1s on the main diagonal and 0s everywhere else. When multiplied by itself, the identity matrix always yields itself as the result. For example, if we have the 2×2 identity matrix I, we can see that:

I * I = [1 0] * [1 0] = [11 + 00 10 + 00] = [1 0] = I

Another example of an idempotent matrix is the projection matrix, which is a square matrix that projects a vector onto a subspace of a vector space. For example, consider the projection matrix P that projects a 3-dimensional vector onto the x-y plane:

P = [1 0 0] [0 1 0]

If we multiply this matrix by itself, we can see that it yields itself as the result:

P * P = [1 0 0] * [1 0 0] [0 1 0] [0 1 0] = [1 0 0] [0 1 0] = P

Therefore, the projection matrix is an idempotent matrix.

There are a few properties of idempotent matrices that are important to understand. First, the rank of an idempotent matrix is always equal to its trace. This means that the number of non-zero elements on the main diagonal of an idempotent matrix is equal to the rank of the matrix. For example, the identity matrix has a rank of 2 because it has 2 non-zero elements on the main diagonal, and the projection matrix has a rank of 2 because it has 2 non-zero elements on the main diagonal.

Second, the inverse of an idempotent matrix is also an idempotent matrix. This means that if we have an idempotent matrix A, we can find its inverse A^-1 by solving the equation A * A^-1 = I, where I is the identity matrix. For example, if we have the projection matrix P, we can find its inverse by solving the equation P * P^-1 = I:

P * P^-1 = I [1 0 0] [a b c] [0 1 0] [d e f] = [1 0] [g h] [0 1]

Solving this equation yields the inverse of the projection matrix P:

P^-1 = [a b c] [d e f] [g h]

Third, idempotent matrices are always diagonalizable. This means that they can be written as a product of a diagonal matrix and an invertible matrix. For example, if we have the projection matrix P, we can write it as the product of the diagonal matrix D and the invertible matrix V:

P = D * V

where D is the diagonal matrix:

D = [1 0 0] [0 1 0]

and V is the invertible matrix:

V = [1 0 0] [0 1 0]

Therefore, idempotent matrices have many useful properties and applications in linear algebra and other areas of mathematics and computer science.