# Diagonal matrix

In matrix algebra, a **diagonal matrix** is a square matrix for which only the entries on the main diagonal can be non-zero, and all the other, off-diagonal, entries are equal to zero.

The sum and product of diagonal matrices are again diagonal, and the diagonal matrices form a subring of the ring of square matrices: indeed for *n*×*n* matrices over a ring *R* this ring is isomorphic to the product ring *R*^{n}.

## Examples

The zero matrix and the identity matrix are diagonal: they are the additive and multiplicative identity respectively of the ring.

## Properties

The diagonal entries are the eigenvalues of a diagonal matrix.

The determinant of a diagonal matrix is the product of the diagonal elements.

A matrix over a field may be transformed into a diagonal matrix by a combination of row and column operations: this is the LDU decomposition.

## Diagonalizable matrix

A **diagonalizable** matrix is a square matrix which is similar to a diagonal matrix: that is, *A* is diagonalizable if there exists an invertible matrix *P* such that is diagonal. The following conditions are equivalent:

*A*is diagonalizable;- The minimal polynomial of
*A*has no repeated roots; *A*is*n*×*n*and has*n*linearly independent eigenvectors.