Matrix analysis: Difference between revisions

Revision as of 01:43, 18 October 2021

Revision as of 22:11, 17 February 2022

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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=978-052-183-940-2|edition=2nd|url=https://books.google.com/books?id=5I5AYeeh0JUC&q=matrix+analysis}}

</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), 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}}

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==Matrix spaces==

The set of all ''m''&thinsp;×&thinsp;''n'' matrices over a [[field (mathematics)|field]] ''F'' denoted in this article ''M''_''mn''(''F'') form a [[vector space]]. Examples of ''F'' include the set of [[rational number]]s <math>\mathbb{Q}</math>, the [[real number]]s <math>\mathbb{R}</math>, and set of [[complex number]]s <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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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 2×22&thinsp;×&thinsp;2 matrices over the field of real numbers, ''M''<submath>M_{22</sub>}(<math>\mathbb{R})</math>), one legitimate basis set of matrices is:

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

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

because any 2×22&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}

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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 [[root of 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]].

[[Properties of polynomial roots|Roots of polynomial]]spolynomials, in this context the eigenvalues, can all be different, or some may be equal (in which case 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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{{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>

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===Frobenius norm===

The '''Frobenius norm''' is analogous to the [[dot product]] of Euclidean vectors; multiply matrix elements entry-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>