Linear and multilinear algebra; matrix theory Definition

Idempotent matrix

Definition

Let $R$ be a ring and let $P$ be an $n \times n$ square matrix over $R$ . The matrix $P$ is idempotent when $P^2 = P$ .

Remarks

The matrices $0$ , $I_n$ and $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ are idempotent. Over a field, an idempotent matrix is the matrix of a projection onto its image along its kernel. If $P$ is idempotent, then so is $I - P$ . Over a field, the eigenvalues of an idempotent matrix are in $\{0, 1\}$ , and its rank equals its trace in characteristic $0$ .

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Matrix definition
Square matrix definition
Idempotent matrix definition

Where this leads

2 pages build on this one.

Tag	`0K7TD9BV2D`
URI	`m2://0K7TD9BV2D`
Kind	definition
Writing	W1 stated
MSC	15A99
Defines	Idempotent matrix
Uses	Square matrix

Idempotent matrix

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by