Compound matrix: Difference between revisions

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In [[linear algebra]], a branch of [[mathematics]], a ('''multiplicative''') '''compound matrix''' is a [[matrix (mathematics)|matrix]] whose entries are all [[minor (linear algebra)|minors]], of a given size, of another matrix.<ref>Horn, Roger A. and Johnson, Charles R., ''Matrix Analysis'', 2nd edition, Cambridge University Press, 2013, {{isbn|978-0-521-54823-6}}, p. 21</ref><ref name=":0">{{Cite journal|last=Muldowney|first=James S.|date=1990|title=Compound matrices and ordinary differential equations|url=http://projecteuclid.org/euclid.rmjm/1181073047|journal=Rocky Mountain Journal of Mathematics|language=en|volume=20|issue=4|pages=857–872|doi=10.1216/rmjm/1181073047|issn=0035-7596|via=|doi-access=free}}</ref> Compound matrices are closely related to [[exterior algebra]]s.

== Definition ==

Let {{math|''A''}} be an {{math|''m'' &thinsp;× &thinsp;''n''}} matrix with [[real number|real]] or [[complex number|complex]] entries.{{efn|The definition, and the purely algebraic part of the theory, of compound matrices requires only that the matrix have entries in a [[commutative ring]]. In this case, the matrix corresponds to a [[module homomorphism|homomorphism]] of [[finitely generated module|finitely generated]] [[free modulesmodule]]s.}} If {{math|''I''}} is a [[subset]] of {{math|{1, ..., ''m''<nowiki>}</nowiki>}} and {{math|''J''}} is a subset of {{math|{1, ..., ''n''<nowiki>}</nowiki>}}, then the '''{{math|(''I'', ''J''&hairsp;)}}-submatrix of {{math|''A''}}''', written {{math|''A''_{''I'', ''J''}}}&hairsp;, is the submatrix formed from {{math|''A''}} by retaining only those rows indexed by {{math|''I''}} and those columns indexed by {{math|''J''}}. If {{math|1=''r'' = ''s''}}, then {{math|det &thinsp;''A''_{''I'', ''J''}}} is the '''{{math|(''I'', ''J''&hairsp;)}}-[[minor (linear algebra)|minor]]''' of {{math|''A''}}.

The '''''r''&hairsp;th compound matrix''' of {{math|''A''}} is a matrix, denoted {{math|''C''_{''r''&thinsp;}(''A'')}}, is defined as follows. If {{math|''r'' > min(''m'', ''n'')}}, then {{math|''C''_{''r''&thinsp;}(''A'')}} is the unique {{math|0 &thinsp;× &thinsp;0}} matrix. Otherwise, {{math|''C''_{''r''&thinsp;}(''A'')}} has size <math display="inline">\binom{m}{r} \!\times\! \binom{n}{r}</math>. Its rows and columns are indexed by {{math|''r''}}-element subsets of {{math|{1, ..., ''m''<nowiki>}</nowiki>}} and {{math|{1, ..., ''n''<nowiki>}</nowiki>}}, respectively, in their lexicographic order. The entry corresponding to subsets {{math|''I''}} and {{math|''J''}} is the minor {{math|det &thinsp;''A''_{''I'', ''J''}}}.

In some applications of compound matrices, the precise ordering of the rows and columns is unimportant. For this reason, some authors do not specify how the rows and columns are to be ordered.<ref>Kung, Rota, and Yan, p. 305.</ref>

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For example, consider the matrix

:<math>A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \end{pmatrix}.</math>

The rows are indexed by {{math|{1, 2, 3<nowiki>}</nowiki>}} and the columns by {{math|{1, 2, 3, 4<nowiki>}</nowiki>}}. Therefore, the rows of {{math|''C''_2&hairsp;(''A'')}} are indexed by the sets

:<math>\{1, 2\} < \{1, 3\} < \{2, 3\}</math>

and the columns are indexed by

:<math>\{1, 2\} < \{1, 3\} < \{1, 4\} < \{2, 3\} < \{2, 4\} < \{3, 4\}.</math>

Using absolute value bars to denote determinants[[determinant]]s, the second compound matrix is

:<math>\begin{align}

C_2(A)

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==Properties==

Let {{math|''c''}} be a scalar, {{math|''A''}} be an {{math|''m'' &thinsp;× &thinsp;''n''}} matrix, and {{math|''B''}} be an {{math|''n'' &thinsp;× &thinsp;''p''}} matrix. IfFor {{math|''k''}} is a positive [[integer]], thenlet {{math|''I''_''k''}} denotesdenote the {{math|''k'' &thinsp;× &thinsp;''k''}} [[identity matrix]]. The [[transpose]] of a matrix {{math|''M''}} will be written {{math|''M''{{i sup|T}}}}, and the [[conjugate transpose]] by {{math|''M''{{i sup|*}}}}. Then:<ref>Horn and Johnson, p. 22.</ref>

* {{math|1=''C''_0&hairsp;(''A'') = ''I''₁}}, a {{math|1 &thinsp;× &thinsp;1}} identity matrix.

* {{math|1=''C''₁(''A'') = ''A''}}.

* {{math|1=''C''_{''r''&thinsp;}(''cA'') = ''c''{{i sup|''r''}}''C''_{''r''&thinsp;}(''A'')}}.

* If {{math|1=rk ''A'' = ''r''}}, then {{math|1=rk C_{''r''&thinsp;}(''A'') = 1}}.

* If {{math|1 &lethinsp;≤ ''r'' &le;≤ ''n''}}, then <math>C_r(I_n) = I_{\binom{n}{r}}</math>.

* If {{math|1 &lethinsp;≤ ''r'' &le;≤ min(''m'', ''n'')}}, then {{math|1=''C''_{''r''&thinsp;}(''A''{{i sup|T}}) = ''C''_{''r''&thinsp;}(''A''){{i sup|T}}}}.

* If {{math|1 &lethinsp;≤ ''r'' &le;≤ min(''m'', ''n'')}}, then {{math|1=''C''_{''r''&thinsp;}(''A''{{i <sup|>*}}</sup>) = ''C''_{''r''&thinsp;}(''A''){{i <sup|>*}}</sup>}}.

* {{math|1=''C''_{''r''&thinsp;}(''AB'') = ''C''_{''r''&thinsp;}(''A'')&hairsp;''C''_{''r''&thinsp;}(''B'')}}.

* ([[Cauchy–Binet formula]]) {{math|1=det ''C''_{''r''&thinsp;}(''AB'') = (det ''C''_{''r''&thinsp;}(''A''))&hairsp;(det ''C''_{''r''&thinsp;}(''B'')})}}.

Assume in addition that {{math|''A''}} is a [[square matrix]] of size {{math|''n''}}. Then:<ref>Horn and Johnson, pp. 22, 93, 147, 233.</ref>

* {{math|1=''C''_{''n''&hairsp;}(''A'') = det &thinsp;''A''}}.

* If {{math|''A''}} has one of the following properties, then so does {{math|''C''_{''r''&thinsp;}(''A'')}}:

** [[Upper triangular]],

** [[Lower triangular]],

** [[Diagonal matrix|Diagonal]],

** [[Orthogonal matrix|Orthogonal]],

** [[Unitary matrix|Unitary]],

** [[Symmetric matrix|Symmetric]],

** [[Hermitian matrix|Hermitian]],

** [[Skew-symmetric matrix|Skew-symmetric]],

** [[Skew-hermitian]],

** [[Positive definite matrix|Positive definite]],

** [[Positive semi-definite matrix|Positive semi-definite]],

** [[Normal matrix|Normal]].

* If {{math|''A''}} is [[invertible matrix|invertible]], then so is {{math|''C''_{''r''&thinsp;}(''A'')}}, and {{math|1=''C''_{''r''&thinsp;}(''A''{{i sup|&minus;1}}) = ''C''_{''r''&thinsp;}(''A''){{i sup|&minus;1−1}}}}.

* (Sylvester–Franke theorem) If {{math|1 &lethinsp;≤ ''r'' &le;≤ ''n''}}, then <math>\det C_r(A) = (\det A)^{\binom{n-1}{r-1}}</math>.<ref name="Tornheim1952">{{cite journal|last1=Tornheim|first1=Leonard|title=The Sylvester–Franke Theorem|journal=The American Mathematical Monthly|volume=59|issue=6|year=1952|pages=389–391|issn=0002-9890|doi=10.2307/2306811|jstor=2306811}}</ref><ref>[[Harley Flanders]] (1953) "A Note on the Sylvester-Franke Theorem", [[American Mathematical Monthly]] 60: 543–5, {{mr|id=0057835}}</ref>

==Relation to exterior powers==

{{see also|Exterior algebra}}

Give {{math|'''R'''^''n''}} the [[canonical basis|standard coordinate basis]] {{math|'''e'''₁, ..., '''e'''_''n''}}. The {{math|''r''}}&hairsp;th exterior power of {{math|'''R'''^''n''}} is the [[vector space]]

:<math>\wedge^r \mathbf{R}^n</math>

whose [[basis (linear algebra)|basis]] consists of the formal symbols

:<math>\mathbf{e}_{i_1} \wedge \dots \wedge \mathbf{e}_{i_r},</math>

where

:<math>i_1 < \dots < i_r.</math>

Suppose that {{math|''A''}} beis an {{math|''m'' &thinsp;× &thinsp;''n''}} matrix. Then {{math|''A''}} corresponds to a [[linear transformation]]

:<math>A \colon \mathbf{R}^n \to \mathbf{R}^m.</math>

Taking the {{math|''r''}}&hairsp;th exterior power of this linear transformation determines a linear transformation

:<math>\wedge^r A \colon \wedge^r \mathbf{R}^n \to \wedge^r \mathbf{R}^m.</math>

The matrix corresponding to this linear transformation (with respect to the above bases of the exterior powers) is {{math|''C''_{''r''&thinsp;}(''A'')}}. Taking exterior powers is a [[functor]], which means that<ref>Joseph P.S. Kung, Gian-Carlo Rota, and [[Catherine Yan|Catherine H. Yan]], ''[[Combinatorics: The Rota Way]]'', Cambridge University Press, 2009, p. 306. {{isbn|9780521883894}}</ref>

:<math>\wedge^r (AB) = (\wedge^r A)(\wedge^r B).</math>

This corresponds to the formula {{math|1=''C''_{''r''&thinsp;}(''AB'') = ''C''_{''r''&thinsp;}(''A'')''C''_{''r''&thinsp;}(''B'')}}. It is closely related to, and is a strengthening of, the [[Cauchy–Binet formula]].

==Relation to adjugate matrices==

{{see also|Adjugate matrix}}

Let {{math|''A''}} be an {{math|''n'' &thinsp;× &thinsp;''n''}} matrix. Recall that its '''{{mathmvar|''r''}}&hairsp;th higher adjugate matrix''' {{math|adj_{''r''&hairsp;}(''A'')}} is the <math display="inline">\binom{n}{r} \!\times\! \binom{n}{r}</math> matrix whose {{math|(''I'', ''J''&hairsp;)}} entry is

:<math>(-1)^{\sigma(I) + \sigma(J)} \det A_{J^c, I^c},</math>

where, for any set {{math|''K''}} of integers, {{math|''σ''(''K'')}} is the sum of the elements of {{math|''K''}}. The '''adjugate''' of {{math|''A''}} is its 1st higher adjugate and is denoted {{math|adj(''A'')}}. The generalized [[Laplace expansion]] formula implies

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If {{math|''A''}} is invertible, then

:<math>\operatorname{adj}_r(A^{-1}) = (\det A)^{-1}C_r(A).</math>

A concrete consequence of this is '''Jacobi's formula''' for the minors of an [[inverse matrix]]:

:<math>\det(A^{-1})_{J^c, I^c} = (-1)^{\sigma(I) + \sigma(J)} \frac{\det A_{I,J}}{\det A}.</math>

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and let {{math|''J''}} denote the ''[[exchange matrix]]'':

:<math>J = \begin{pmatrix} & & 1 \\ & \cdots & \\ 1 & & \end{pmatrix}.</math>

Then '''Jacobi's theorem''' states that the {{math|''r''}}&hairsp;th higher adjugate matrix is:<ref name="NambiarSreevalsan2001">{{cite journal|last1=Nambiar|first1=K.K.|last2=Sreevalsan|first2=S.|title=Compound matrices and three celebrated theorems|journal=Mathematical and Computer Modelling|volume=34|issue=3–4|year=2001|pages=251–255|issn=0895-7177|doi=10.1016/S0895-7177(01)00058-9|doi-access=free}}</ref><ref name="Price1947">{{cite journal|last1=Price|first1=G. B.|authorlink=G. B. Price|title=Some Identities in the Theory of Determinants|journal=The American Mathematical Monthly|volume=54|issue=2|year=1947|pages=75–90|issn=0002-9890|doi=10.2307/2304856|jstor=2304856}}</ref>

:<math>\operatorname{adj}_r(A) = JC_{n-r}(SAS)^TJ.</math>

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The computation of compound matrices appears in a wide array of problems.<ref>{{cite techreport|first=Boutin|last=D.L.|author2=R.F. Gleeson|author3=R.M. Williams|title=Wedge Theory / Compound Matrices: Properties and Applications.|institution=Office of Naval Research|url=https://apps.dtic.mil/sti/pdfs/ADA320264.pdf|year=1996|number=NAWCADPAX–96-220-TR}}</ref><ref name=":0" />

Compound and adjugate matrices appear when computing determinants of [[linear combinationscombination]]s of matrices. It is elementary to check that, if {{math|''A''}} and {{math|''B''}} are {{math|''n'' &thinsp;× &thinsp;''n''}} matrices, then

:<math>\det(sA + tB) = C_n\!\left(\begin{bmatrix} sA & I_n \end{bmatrix}\right)C_n\!\left(\begin{bmatrix} I_n \\ tB \end{bmatrix}\right).</math>

It is also true that:<ref>{{Cite journal|last=Prells|first=Uwe|last2=Friswell|first2=Michael I.|last3=Garvey|first3=Seamus D.|date=2003-02-08|title=Use of geometric algebra: compound matrices and the determinant of the sum of two matrices|url=http://rspa.royalsocietypublishing.org/content/459/2030/273|journal=Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences|language=en|volume=459|issue=2030|pages=273–285|doi=10.1098/rspa.2002.1040|issn=1364-5021}}</ref><ref>Horn and Johnson, p. 29</ref>

:<math>\det(sA + tB) = \sum_{r=0}^n s^r t^{n-r} \operatorname{tr}(\operatorname{adj}_r(A)C_r(B)).</math>

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== Numerical computation ==

In general, the computation of compound matrices is non -effective due to its high complexity. Nonetheless, there isare some efficient algorithms available for real matrices with special structuresstructure.<ref>{{Cite journal|last=Kravvaritis|first=Christos|last2=Mitrouli|first2=Marilena|date=2009-02-01|title=Compound matrices: properties, numerical issues and analytical computations|url=http://users.uoa.gr/~mmitroul/mmitroulweb/numalg09.pdf|journal=Numerical Algorithms|language=en|volume=50|issue=2|pages=155|doi=10.1007/s11075-008-9222-7|issn=1017-1398}}</ref>

==Notes==