Diagonal matrix: Difference between revisions

Revision as of 07:14, 29 August 2021

Revision as of 16:41, 29 August 2021

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

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

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 non-zerononzero. An example of a 2×2 diagonal matrix is <math>\left[\begin{smallmatrix}

3 & 0 \\

0 & 2 \end{smallmatrix}\right]</math>, while an example of a 3×3 diagonal matrix is<math>

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:{{nowrap|diag(''a''₁, ..., ''a''_''n'')}} {{nowrap|diag(''b''₁, ..., ''b''_''n'')}} = {{nowrap|diag(''a''₁''b''₁, ..., ''a''_''n''''b''_''n'')}}.

The diagonal matrix {{nowrap|diag(''a''₁, ..., ''a''_''n'')}} is [[invertible matrix|invertible]] [[if and only if]] the entries ''a''₁, ..., ''a''_''n'' are all non-zerononzero. In this case, we have

:{{nowrap|diag(''a''₁, ..., ''a''_''n'')⁻¹}} = {{nowrap|diag(''a''₁⁻¹, ..., ''a''_''n''⁻¹)}}.