Diagonal matrix

This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices

Definition

A diagonal matrix is a square matrix for which all the off-diagonal entries are zero, or equivalently, all nonzero entries are on the diagonal. Note that it is also possible that some (or even all) the diagonal entries are zero.

Encoding

A matrix that is known to be diagonal may simply be encoded using an ordered list of its diagonal entries. For a $n\times n$ matrix, this requires space for $n$ entries (in contrast with space for $n^{2}$ entries for an arbitrary square matrix).

Matrix operations

Unless otherwise specified, diagonal matrices are $n\times n$ diagonal matrices.

Matrix operation	Arithmetic complexity: number of additions	Arithmetic complexity: number of multiplications
Computation of the trace	$n-1$	0
Computation of the determinant	0	$n-1$
Addition of two such matrices	$n$	0
Multiplication of two such matrices	0	$n$
Multiplication by a $n\times p$ rectangular matrix (we allow $n=p$)	0	$np$