Diagonal matrix: Difference between revisions

Revision as of 06:01, 13 November 2024

Latest revision as of 23:43, 27 June 2025

(5 intermediate revisions by 4 users not shown)

Line 1:

{{Use American English|date = March 2019}}

{{Short description|Matrix whose only nonzero elements are on its main diagonal}}

{{More footnotes needed|date=June 2025}}

In [[linear algebra]], a '''diagonal matrix''' is a [[matrix (mathematics)|matrix]] in which the entries outside the [[main diagonal]] are all zero; the term usually refers to [[square matrices]]. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is <math>\left[\begin{smallmatrix}

Line 16 ⟶ 17:

==Definition==

As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix {{math|1='''D''' = (''d''_''i'',''j'')}} with {{mvar|n}} columns and {{mvar|n}} rows is diagonal if

<math display="block">\forall i,j \in \{1, 2, \ldots, n\}, i \ne j \implies d_{i,j} = 0.</math>

Line 48 ⟶ 49:

==Vector-to-matrix diag operator==

A diagonal matrix {{math|'''D'''}} can be constructed from a vector <math>\mathbf{a} = \begin{bmatrix}a_1 & \dotsmdots & a_n\end{bmatrix}^\textsf{T}</math> using the <math>\operatorname{diag}</math> operator:

<math display="block">
\mathbf{D} = \operatorname{diag}(a_1, \dots, a_n).
</math>

This may be written more compactly as <math>\mathbf{D} = \operatorname{diag}(\mathbf{a})</math>.

The same operator is also used to represent [[Block matrix#Block diagonal matrices|block diagonal matrices]] as <math> \mathbf{A} = \operatorname{diag}(\mathbf A_1, \dots, \mathbf A_n)</math> where each argument {{math|'''A'''{{sub|''i''}}}} is a matrix.

The {{math|diag}} operator may be written as:

<math display="block">
\operatorname{diag}(\mathbf{a}) = \left(\mathbf{a} \mathbf{1}^\textsf{T}\right) \circ \mathbf{I},
</math>

where <math>\circ</math> represents the [[Hadamard product (matrices)|Hadamard product]], and {{math|'''1'''}} is a constant vector with elements 1.

==Matrix-to-vector diag operator==

The inverse matrix-to-vector {{math|diag}} operator is sometimes denoted by the identically named <math>\operatorname{diag}(\mathbf{D}) = \begin{bmatrix}a_1 & \dotsmdots & a_n\end{bmatrix}^\textsf{T},</math> where the argument is now a matrix, and the result is a vector of its diagonal entries.

The following property holds:

<math display="block">
\operatorname{diag}(\mathbf{A}\mathbf{B}) = \sum_j \left(\mathbf{A} \circ \mathbf{B}^\textsf{T}\right)_{ij} = \left( \mathbf{A} \circ \mathbf{B}^\textsf{T} \right) \mathbf{1} .
</math>

== Scalar matrix ==

{{Confusing|section|reason=many sentences use incorrect, awkward grammar and should be reworded to make sense|date=February 2021}}

<!-- Linked from [[Scalar matrix]] and [[Scalar transformation]] -->

A diagonal matrix with equal diagonal entries is a '''scalar matrix'''; that is, a scalar multiple {{mvar|λ}} of the [[identity matrix]] {{math|'''I'''}}. Its effect on a [[vector (mathematics and physics)|vector]] is [[scalar multiplication]] by {{mvar|λ}}. For example, a 3×3 scalar matrix has the form: