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 main 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:

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

Note that for an actual diagonal matrix, the symbols $a_{11},a_{22},\dots ,a_{nn}$ will be replaced by values in the underlying set or ring the matrix is over.

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).

Explicitly, the ordered list $(a_{11},a_{22},\dots ,a_{nn})$ can be used to describe the diagonal matrix $A=(a_{ij})$. Further:

• This encoding rule takes $n$ times the space needed to store a single entry.
• It is easy to convert back and forth between this encoding rule and the matrix description. In one direction, given this encoding rule, and numbers $i,j\in \{1,2,\dots ,n\}$, we can easily determine $a_{ij}$. In the reverse direction, given a matrix encoded in the usual manner, we can look up the diagonal entries and construct the encoding. (To confirm that the matrix is diagonal, we need to look at all non-diagonal matrix entries).

Matrix operations

Invariant computation

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

Matrix operation	Description in terms of diagonal encoding: we use $(a_{11},a_{22},\dots ,a_{nn})$ when there is only one matrix	Arithmetic complexity: number of additions	Arithmetic complexity: number of multiplications	Number of inverse computations
Computation of the trace	$\sum _{i=1}^{n}a_{ii}$	$n-1$	0	0
Computation of the determinant	$\prod _{i=1}^{n}a_{ii}$	0	$n-1$	0

Operations

Matrix operation	Description in terms of diagonal encoding: we denote matrices as $(a_{11},a_{22},\dots ,a_{nn})$ and (if the operation is binary)$(b_{11},b_{22},\dots ,b_{nn})$	Arithmetic complexity: number of additions	Arithmetic complexity: number of multiplications	Number of inverse computations	Set of diagonal matrices closed under operation?
Addition of two diagonal matrices	$(a_{11},a_{22},\dots ,a_{nn})+(b_{11},b_{22},\dots ,b_{nn})=(a_{11}+b_{11},a_{22}+b_{22},\dots ,a_{nn}+b_{nn})$	$n$	0	0	Yes
Multiplication of two diagonal matrices	$(a_{11},a_{22},\dots ,a_{nn})(b_{11},b_{22},\dots ,b_{nn})=(a_{11}b_{11},a_{22}b_{22},\dots ,a_{nn}b_{nn})$	0	$n$	0	Yes
Computation of the inverse matrix	$(a_{11},a_{22},\dots ,a_{nn})^{-1}=(a_{11}^{-1},a_{22}^{-1},\dots ,a_{nn}^{-1})$ (inverse does not exist if any $a_{ii}=0$	0	0	$n$	Yes, though not all diagonal matrices are invertible

The set of diagonal matrices has an algebraic structure of a ring. Explicitly, if the matrix entries are over a commutative unital ring $R$, then the set of diagonal matrices form a commutative unital ring $R^{n}$: an external direct product of $n$ copies of $R$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
scalar matrix	diagonal matrix where all the diagonal entries are equal	obvious	A matrix such as ${\begin{pmatrix}1&0\\0&0\\\end{pmatrix}}$ is diagonal but not scalar.	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
diagonalizable matrix	is similar to a diagonal matrix. Note that the notion of diagonalizability depends on the ring we are considering matrices over, so a given matrix may be diagonalizable in one ring but not in a smaller ring.	(obvious)	diagonalizable not implies diagonal	|FULL LIST, MORE INFO
upper triangular matrix	all entries below the main diagonal are zero	(obvious)	The matrix ${\begin{pmatrix}1&1\\0&1\\\end{pmatrix}}$ is upper triangular but not diagonal	|FULL LIST, MORE INFO
lower triangular matrix	all entries above the main diagonal are zero	(obvious)	The matrix ${\begin{pmatrix}1&0\\1&1\\\end{pmatrix}}$ is upper triangular but not diagonal	|FULL LIST, MORE INFO
tridiagonal matrix	all nonzero entries are on the main diagonal, the first superdiagonal, or the first subdiagonal	(obvious)	any non-diagonal $2\times 2$ matrix is tridiagonal	|FULL LIST, MORE INFO
symmetric matrix	equals its matrix transpose, or equivalently, $a_{ij}=a_{ji}$ for all $i,j$	diagonal implies symmetric	The matrix ${\begin{pmatrix}0&1\\1&0\\\end{pmatrix}}$ is a counterexample.	|FULL LIST, MORE INFO

Non-binary properties where diagonal matrices are exemplars

Attribute	Meaning	Name for matrices that do "well" on this criterion	Value for diagonal matrix
sparsity of a matrix	fraction of matrix entries that are zero	sparse matrix refers to a matrix whose sparsity is high (preferably close to 1). A family of matrices of size going to infinity is sparse if the sparsity of the family approaches 1.	At least $1-(1/n)$. In the limit as $n\to \infty$, this approaches 1.
bandwidth of a matrix	the number of adjacent diagonals that are needed to cover all nonzero entries	band matrix refers to a matrix of low bandwidth.	At most 1 (0 only if it is the zero matrix, 1 otherwise). The left and right half bandwidths are both 0.