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 Rⁿ.

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 $P^{-1}AP$ 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.