Matrix analysis: Difference between revisions

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{{Short description|Study of matrices and their algebraic properties}}

In [[mathematics]], particularly in [[linear algebra]] and applications, '''matrix analysis''' is the study of [[matrix (mathematics)|matrices]] and their algebraic properties.<ref>{{cite book|title=Matrix Analysis|author=R. A. Horn, C. R. Johnson|year=2012|publisher=Cambridge University Press|isbn=978-052-183-940-82|edition=2nd|url=httphttps://books.google.co.ukcom/books?id=5I5AYeeh0JUC&printsec=frontcover&dqq=matrix+analysis&hl=en&sa=X&ei=bC91Ut2rCPKO7Qbh8IBI&redir_esc=y#v=onepage&q=matrix%20analysis&f=false}}

</ref> Some particular topics out of many include; operations defined on matrices (such as [[matrix addition]], [[matrix multiplication]] and operations derived from these), functions of matrices (such as [[matrix exponentiation]] and [[matrix logarithm]], and even [[sine]]ssines and cosines]] etc. of matrices), <!---[[matrix calculus]] (matrix [[derivative|differentiation]] and [[integral (mathematics)|integration]]), --->and the [[eigenvalue]]s of matrices ([[eigendecomposition of a matrix]], [[eigenvalue perturbation]] theory).<ref>{{cite book|title=Functions of Matrices: Theory and Computation|author=N. J. Higham|year=2000 |publisher=SIAM|isbn=089-871-777-9|url=https://books.google.com/books?id=S6gpNn1JmbgC&q=matrix+functions}}

</ref>

==Matrix spaces==

The set of all ''m''&thinsp;×&thinsp;''n'' matrices over a number [[field (mathematics)|field]] ''F'' denoted in this article ''M''_''mn''(''F'') form a [[vector space]]. Examples of ''F'' include the set of [[integerrational number]]s ℤ<math>\mathbb{Q}</math>, the [[real number]]s ℝ<math>\mathbb{R}</math>, and set of [[complex number]]s ℂ<math>\mathbb{C}</math>. The spaces ''M''_''mn''(''F'') and ''M''_''pq''(''F'') are different spaces if ''m'' and ''p'' are unequal, and if ''n'' and ''q'' are unequal; for instance ''M''₃₂(''F'') ≠ ''M''₂₃(''F''). Two ''m''&thinsp;×&thinsp;''n'' matrices '''A''' and '''B''' in ''M''_''mn''(''F'') can be added together to form another matrix in the space ''M''_''mn''(''F''):

:<math>\mathbf{A},\mathbf{B} \in M_{mn}(F)\,,\quad \mathbf{A} + \mathbf{B} \in M_{mn}(F) </math>

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:<math>\alpha \mathbf{A} + \beta\mathbf{B} \in M_{mn}(F) </math>

where ''α'' and ''β'' are numbers in ''F''.

Any matrix can be expressed as a linear combination of basis matrices, which play the role of the [[basis vector]]s for the matrix space. For example, for the set of 2&thinsp;×&thinsp;2 matrices over the field of real numbers, <math>M_{22}(\mathbb{R})</math>, one legitimate basis set of matrices is:

:<math>\begin{pmatrix}1&0\\0&0\end{pmatrix}\,,\quad

\begin{pmatrix}0&1\\0&0\end{pmatrix}\,,\quad

\begin{pmatrix}0&0\\1&0\end{pmatrix}\,,\quad

\begin{pmatrix}0&0\\0&1\end{pmatrix}\,,</math>

because any 2&thinsp;×&thinsp;2 matrix can be expressed as:

:<math>\begin{pmatrix}a&b\\c&d\end{pmatrix}=a \begin{pmatrix}1&0\\0&0\end{pmatrix}

+b\begin{pmatrix}0&1\\0&0\end{pmatrix}

+c\begin{pmatrix}0&0\\1&0\end{pmatrix}

+d\begin{pmatrix}0&0\\0&1\end{pmatrix}\,,</math>

where ''a'', ''b'', ''c'',''d'' are all real numbers. This idea applies to other fields and matrices of higher dimensions.

==Determinants==

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{{main|Determinant}}

The '''determinant''' of a [[square matrix]] is an important property. The determinant indicates if a matrix is [[invertible]] (i.e. the [[inverse matrix|inverse of a matrix]] exists when the determinant is nonzero). Determinants are used for finding eigenvalues of matrices (see below), and for solving a [[system of linear equations]] (see [[Cramer's rule]]).

==Eigenvalues and eigenvectors of matrices==

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===Definitions===

An ''n''&thinsp;×&thinsp;''n'' matrix '''A''' has '''eigenvectors''' '''x''' and '''eigenvalues''' ''λ'' defined by the relation:

:<math>\mathbf{A}\mathbf{x} = \lambda \mathbf{x}</math>

In words, the [[matrix multiplication]] of '''A''' followed by an eigenvector '''x''' (here an ''n''-dimensional [[column matrix]]), is the same as multiplying the eigenvector by the eigenvalue. For an ''n''&thinsp;×&thinsp;''n'' matrix, there are ''n'' eigenvalues. The eigenvalues are the solutions[[root toof a polynomial|roots]] of the [[characteristic polynomial]]:

:<math>p_\mathbf{A}(\lambda) = \det(\mathbf{A} - \lambda \mathbf{I}) = 0</math>

where '''I''' is the ''n''&thinsp;×&thinsp;''n'' [[identity matrix]].

Roots of polynomials, in this context the eigenvalues, can all be different, or some may be equal (in which case the eigenvalue has [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicity]], the number of times an eigenvalue occurs). After solving for the eigenvalues, the eigenvectors corresponding to the eigenvalues can be found by the defining equation.

===Perturbations of eigenvalues===

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==Matrix similarity==

{{main|Matrix similarity|Change of basis}}

Two ''n''&thinsp;×&thinsp;''n'' matrices '''A''' and '''B''' are similar if they are related by a '''similarity transformation''':

:<math>\mathbf{B} = \mathbf{P}\mathbf{A}\mathbf{P}^{-1}</math>

The matrix '''P''' is called a '''similarity matrix''', and is necessarily [[matrix inverse|invertible]].

===Unitary similarity===

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For all matrices '''A''' and '''B''' in ''M''_''mn''(''F''), and all numbers ''α'' in ''F'', a matrix norm, delimited by double vertical bars || ... ||, fulfills:<ref group="note">Some authors, e.g. Horn and Johnson, use triple vertical bars instead of double: |||'''A'''|||.</ref>

*[[Positive definiteNonnegative]]:

::<math>\| \mathbf{A} \| \ge 0</math>

:with equality only for '''A''' = '''0''', the [[zero matrix]].

*[[Scalar multiplication]]:

::<math>\|\alpha \mathbf{A}\|=|\alpha| \|\mathbf{A}\|</math>

*The [[triangular inequality]]:

::<math>\|\mathbf{A}+\mathbf{B}\| \leq \|\mathbf{A}\|+\|\mathbf{B}\|</math>

===Frobenius norm===

The '''Frobenius norm''' is analogous to the [[dot product]] of Euclidean vectors; square everymultiply matrix elementelements andentry-wise, add up the results, then take the positive [[square root]]:

:<math>\|\mathbf{A}\| = \sqrt{\mathbf{A}:\mathbf{A}} = \sqrt{\sum_{i=1}^m \sum_{j=1}^n (A_{ij})^2}</math>

It is defined for matrices of any dimension (i.e. no restriction to square matrices).

==Positive definite and semidefinite matrices ==

{{main|Positive definite matrix}}

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{{main|Function (mathematics)}}

Matrix elements are not restricted to constant numbers, they can be [[mathematical variable]]s.

===Functions of matrices ===

A functions of a matrix takes in a matrix, and return something else (a number, vector, matrix, etc...).

===Matrix-valued functions ===

A matrix valued function takes in something (a number, vector, matrix, etc...) and returns a matrix.

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<!---rather than simply deleting, please include these in the article somewhere wherever relevant after the real content is written--->

===Other branches of analysis===

*[[Mathematical analysis]]

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*[[Matrix calculus]]

*[[Numerical analysis]]

*[[Determinant]]

===Other concepts of linear algebra===

*[[Matrix similarity]]

*[[Tensor product]]

*[[Spectrum of an operator]]

*[[Matrix geometrical series]]

===Types of matrix===

*[[Orthogonal matrix]], [[unitary matrix]]

*[[Symmetric matrix]], [[antisymmetric matrix]]

*[[Stochastic matrix]]

===Matrix functions===

*[[Matrix similaritypolynomial]]

*[[Matrix exponential]]

==Footnotes==

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===Further reading===

*{{cite book|title=FunctionsMatrix ofAnalysis Matrices:and Applied Linear Algebra TheoryBook and ComputationSolutions Manual|author=NC. J. HighamMeyer|year=2000 |publisher=SIAM|isbn=089-871-777454-90|volume=2|url=httphttps://books.google.co.ukcom/books?id=S6gpNn1JmbgCZg4M0iFlbGcC&printsecq=frontcover&dq=matrixMatrix+functions&hl=en&sa=X&ei=_x1-UqDLE4qV7Qa5s4DAAg&redir_esc=y#v=onepage&q=matrix%20functions&f=falseAnalysis}}

*{{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=978-038-733-195-69|series=[[Undergraduate Texts in Mathematics]]|url=httphttps://books.google.co.ukcom/books?id=8qwTb9P-iW8C&printsec=frontcover&dqq=Matrix+Analysis&hl=en&sa=X&ei=SCd1UryWD_LG7Aag_4HwBg&ved=0CGQQ6AEwCA#v=onepage&q=Matrix%20Analysis&f=false}}

*{{cite book|title=Matrix Analysis and Applied Linear Algebra Book and Solutions Manual|author=C.Rajendra MeyerBhatia|year=20001997|volume=169|series=Matrix Analysis Series|publisher=SIAMSpringer|isbn=089038-871794-454846-0|volume=2|series=Matrix Analysis and Applied Linear Algebra5|url=httphttps://books.google.co.ukcom/books?id=Zg4M0iFlbGcCF4hRy1F1M6QC&printsecq=frontcover&dq=Matrixmatrix+Analysis&hl=en&sa=X&ei=SCd1UryWD_LG7Aag_4HwBg&ved=0CGoQ6AEwCQ#v=onepage&q=Matrix%20Analysis&f=falseanalysis}}

*{{cite book|title=Computational Matrix Analysis|author=Alan J. Laub|year=2012|publisher=SIAM|isbn=978-161-197-221-34|url=httphttps://books.google.co.ukcom/books?id=RJBZBuHpVjEC&printsec=frontcover&dqq=Matrix+Analysis&hl=en&sa=X&ei=Iyl1UtCuEIbm7Abc4YHoCg&ved=0CDAQ6AEwADgK#v=onepage&q=Matrix%20Analysis&f=false}}

*{{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=038-733-195-6|series=Undergraduate Texts in Mathematics|url=http://books.google.co.uk/books?id=8qwTb9P-iW8C&printsec=frontcover&dq=Matrix+Analysis&hl=en&sa=X&ei=SCd1UryWD_LG7Aag_4HwBg&ved=0CGQQ6AEwCA#v=onepage&q=Matrix%20Analysis&f=false}}

*{{cite book|title=Matrix Analysis|author=Rajendra Bhatia|year=1997|volume=169|series=Matrix Analysis Series|publisher=Springer|isbn=038-794-846-5|url=http://books.google.co.uk/books?id=F4hRy1F1M6QC&printsec=frontcover&dq=matrix+analysis&hl=en&sa=X&ei=_SR1UpbnNarA7AaPjIHIDA&redir_esc=y#v=onepage&q=matrix%20analysis&f=false}}

*{{cite book|title=Computational Matrix Analysis|author=Alan J. Laub|year=2012|publisher=SIAM|isbn=161-197-221-3|url=http://books.google.co.uk/books?id=RJBZBuHpVjEC&printsec=frontcover&dq=Matrix+Analysis&hl=en&sa=X&ei=Iyl1UtCuEIbm7Abc4YHoCg&ved=0CDAQ6AEwADgK#v=onepage&q=Matrix%20Analysis&f=false}}

{{mathematics-stub}}

[[Category:Linear algebra]]

[[Category:Matrices (mathematics)]]

[[Category:Numerical analysis]]