Diagonal matrix: Difference between revisions

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{{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}

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==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: