Diagonal matrix: Difference between revisions

Revision as of 14:04, 17 November 2021

Revision as of 18:12, 17 November 2021

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In the remainder of this article we will consider only square diagonal matrices, and refer to them simply as "diagonal matrices".

==Vector-to-matrix diag operator==

A diagonal matrix <math>D</math> can be constructed from a vector <math>\mathbf{a} = \begin{bmatrix}a_1 & \dotsm & a_n\end{bmatrix}^\textsf{T}</math> using the <math>\operatorname{diag}</math> operator:

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

The inverse of this operator is sometimes denoted by the identically named <math>\operatorname{diag}(D) = \begin{bmatrix}a_1 & \dotsm & 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 same operator name is also used to represent [[Block matrix#Block diagonal matrices|block diagonal matrices]] as <math> A = \operatorname{diag}(A_1, \dots, A_n)</math> where each argument <math>A_i</math> is a matrix.

The <math>\operatorname{diag}</math> operator may be written as:

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where <math>\circ</math> represents the [[Hadamard product]] and <math>\mathbf{1}</math> is a constant vector with elements 1.

==Matrix-to-vector diag operator==

The inverse matrix-to-vector <math>\operatorname{diag}</math> operator is sometimes denoted by the identically named <math>\operatorname{diag}(D) = \begin{bmatrix}a_1 & \dotsm & 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>\operatorname{diag}(AB) = \sum_j (A \circ B^T)_{ij} </math>

== Scalar matrix ==