An idempotent matrix is a square matrix which when multiplied by itself, gives the resultant matrix as itself. In other words, a matrix P is called idempotent if P2 = P.
Idempotent matrix Examples
The examples of 2 x 2 idempotent matrices are
1. \[\begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix}\]
2. \[\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}\]
This second matrix is the identity matrix.
The examples of 3 x 3 idempotent matrices are
1. \[\begin{bmatrix} 2 &-2 &-4 \\ -1& 3 &4 \\ 1&-2 & -3 \end{bmatrix}\]
2. \[\begin{bmatrix} 1 &0 &0 \\ 0& 1 &0 \\ 0&0 & 1 \end{bmatrix}\]
How do you know if a matrix is idempotent?
It is easy to check whether a matrix is idempotent or not. Simply, check that square of a matrix is the matrix itself or not i.e. P2 = P, where P is a matrix. If this condition is satisfied then the matrix is idempotent. If the condition is not satisfied then the matrix is not idempotent.
Matrix one is Idempotent but it is not symmetric. I believe it projects arbitrary vectors onto the span of its column space. But what does it project along?