Diagonal matrix: Difference between revisions

Revision as of 06:02, 13 November 2024

Revision as of 23:16, 23 March 2025

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==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==