Diagonal matrix: Difference between revisions

Revision as of 03:34, 23 November 2023

Revision as of 23:13, 1 December 2023

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==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>

However, the main diagonal entries are unrestricted.

The term ''diagonal matrix'' may sometimes refer to a '''{{visible anchor|rectangular diagonal matrix}}''', which is an ''{{mvar|m''}}-by-''{{mvar|n''}} matrix with all the entries not of the form {{math|''d''_''i'',''i''}} being zero. For example:

:<math display=block>\begin{bmatrix}

1 & 0 & 0\\

0 & 4 & 0\\

0 & 0 & -3\\

0 & 0 & 0\\

\end{bmatrix}</math> \quad \text{or} \quad <math>\begin{bmatrix}

1 & 0 & 0 & 0 & 0\\

0 & 4 & 0& 0 & 0\\

0 & 0 & -3& 0 & 0

\end{bmatrix}</math>

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== Matrix operations ==

The operations of matrix addition and [[matrix multiplication]] are especially simple for diagonal matrices. Write {{math|diag(''a''₁, ..., ''a''_''n'''')}} for a diagonal matrix whose diagonal entries starting in the upper left corner are {{math|''a''₁, ..., ''a''_''n''''}}. Then, for [[matrix addition|addition]], we have

<math display=block>

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

\operatorname{diag}(a_1,\, \ldots,\, a_n) + \operatorname{diag}(b_1,\, \ldots,\, b_n) = \operatorname{diag}(a_1 + b_1,\, \ldots,\, a_n + b_n)</math>

and for [[matrix multiplication]],

<math display=block>\operatorname{diag}(a_1,\, \ldots,\, a_n) \operatorname{diag}(b_1,\, \ldots,\, b_n) = \operatorname{diag}(a_1 b_1,\, \ldots,\, a_n b_n).</math>

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

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

<math display=block>\operatorname{diag}(a_1,\, \ldots,\, a_n)^{-1} = \operatorname{diag}(a_1^{-1},\, \ldots,\, a_n^{-1}).</math>

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

In particular, the diagonal matrices form a [[subring]] of the ring of all ''{{mvar|n''}}-by-''{{mvar|n''}} matrices.

Multiplying an ''{{mvar|n''}}-by-''{{mvar|n''}} matrix {{mvar|A}} from the ''left'' with {{math|diag(''a''₁, ..., ''a''_''n'''')}} amounts to multiplying the {{mvar|i}}-th ''row'' of {{mvar|A}} by {{mathmvar|''a''_''i''}} for all {{mvar|i}}; multiplying the matrix {{mvar|A}} from the ''right'' with {{math|diag(''a''₁, ..., ''a''_''n'''')}} amounts to multiplying the {{mvar|i}}-th ''column'' of {{mvar|A}} by {{mathmvar|''a''_''i''}} for all {{mvar|i}}.

== Operator matrix in eigenbasis ==

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** A matrix is diagonal if and only if it is both [[triangular matrix|upper-]] and [[triangular matrix|lower-triangular]].

** A diagonal matrix is [[symmetric matrix|symmetric]].

* The [[identity matrix]] ''{{mvar|I''_''n''}} and [[zero matrix]] are diagonal.

* A 1×1 matrix is always diagonal.

* The square of a 2×2 matrix with zero [[trace (linear algebra)|trace]] is always diagonal.

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Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is typically desirable to represent a given matrix or [[linear operator|linear map]] by a diagonal matrix.

In fact, a given ''{{mvar|n''}}-by-''{{mvar|n''}} matrix {{mvar|A}} is [[similar matrix|similar]] to a diagonal matrix (meaning that there is a matrix {{mvar|X}} such that {{math|''X''⁻¹''AX''}} is diagonal) if and only if it has {{mvar|n}} [[linearly independent]] eigenvectors. Such matrices are said to be [[diagonalizable matrix|diagonalizable]].

Over the [[field (mathematics)|field]] of [[real number|real]] or [[complex number|complex]] numbers, more is true. The [[spectral theorem]] says that every [[normal matrix]] is [[matrix similarity|unitarily similar]] to a diagonal matrix (if {{math|1=''AA''^∗ = ''A''^∗''A''}} then there exists a [[unitary matrix]] {{mvar|U}} such that {{math|''UAU''^∗}} is diagonal). Furthermore, the [[singular value decomposition]] implies that for any matrix {{mvar|A}}, there exist unitary matrices {{mvar|U}} and {{mvar|V}} such that {{math|''U^∗AV''}} is diagonal with positive entries.