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

Verbal 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.

Algebraic description

Suppose $n$ is a positive integer. Suppose $A=(a_{ij})_{1\leq i\leq n,1\leq j\leq n}$ is a $n\times n$ matrix. We say that $A$ is a diagonal matrix if the following holds:

$a_{ij}=0\ \forall i\neq j\in \{1,2,\dots ,n\}$

In general, a $n\times n$ diagonal matrix has the following appearance:

${\begin{pmatrix}a_{11}&0&\dots &0\\0&a_{22}&\dots &0\\\cdot &\cdot &\dots &\cdot \\0&0&\dots &a_{nn}\\\end{pmatrix}}$

Small cases

$n$	General description of $n\times n$ diagonal matrix	Number of free parameters (equals $n$)
1	${\begin{pmatrix}a_{11}\end{pmatrix}}$	1
2	${\begin{pmatrix}a_{11}&0\\0&a_{22}\\\end{pmatrix}}$	2
3	${\begin{pmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\\\end{pmatrix}}$	3 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 $np$