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.
Contents
show
Idempotent matrix Examples
The examples of 2 x 2 idempotent matrices are
1.
![\[\begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-f3f01249119386e5e79dd1573a478a39_l3.png "Rendered by QuickLaTeX.com")
2.
![\[\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-681e15696fbb1972fc9a491b4c8c323b_l3.png "Rendered by QuickLaTeX.com")
This second matrix is the identity matrix.
The examples of 3 x 3 idempotent matrices are
1.
![\[\begin{bmatrix} 2 &-2 &-4 \\ -1& 3 &4 \\ 1&-2 & -3 \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-b2f9315fa39344573998f673bfc03a78_l3.png "Rendered by QuickLaTeX.com")
2.
![\[\begin{bmatrix} 1 &0 &0 \\ 0& 1 &0 \\ 0&0 & 1 \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-451e54c66ca28ee56656d115019ec490_l3.png "Rendered by QuickLaTeX.com")
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?