Matrix analysis

The set of all m × n matrices over a field F denoted in this article M_mn(F) form a vector space. Examples of F include the set of rational numbers $\backslash mathbb\{Q\}$, the real numbers $\backslash mathbb\{R\}$, and set of complex numbers $\backslash mathbb\{C\}$. 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 M₃₂(F) ≠ M₂₃(F). Two m × n matrices A and B in M_mn(F) can be added together to form another matrix in the space M_mn(F):

:$\backslash mathbf\{A\},\backslash mathbf\{B\}\; \backslash in\; M\_\{mn\}(F)\backslash ,,\backslash quad\; \backslash mathbf\{A\}\; +\; \backslash mathbf\{B\}\; \backslash in\; M\_\{mn\}(F)$

and multiplied by a α in F, to obtain another matrix in M_mn(F):

:$\backslash alpha\; \backslash in\; F\; \backslash ,,\backslash quad\; \backslash alpha\; \backslash mathbf\{A\}\; \backslash in\; M\_\{mn\}(F)$

Combining these two properties, a linear combination of matrices A and B are in M_mn(F) is another matrix in M_mn(F):

:$\backslash alpha\; \backslash mathbf\{A\}\; +\; \backslash beta\backslash mathbf\{B\}\; \backslash in\; M\_\{mn\}(F)$

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 vectors for the matrix space. For example, for the set of 2 × 2 matrices over the field of real numbers, $M\_\{22\}(\backslash mathbb\{R\})$, one legitimate basis set of matrices is:

:

\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}\,,

because any 2 × 2 matrix can be expressed as:

:

+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}\,,

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

{{main|Determinant}}

The determinant of a square matrix is an important property. The determinant indicates if a matrix is invertible (i.e. the 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).

{{main|Eigenvalues and eigenvectors}}

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

:$\backslash mathbf\{A\}\backslash mathbf\{x\}\; =\; \backslash lambda\; \backslash mathbf\{x\}$

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 × n matrix, there are n eigenvalues. The eigenvalues are the roots of the characteristic polynomial:

:$p\_\backslash mathbf\{A\}(\backslash lambda)\; =\; \backslash det(\backslash mathbf\{A\}\; -\; \backslash lambda\; \backslash mathbf\{I\})\; =\; 0$

where I is the n × n identity matrix.

Roots of polynomials, in this context the eigenvalues, can all be different, or some may be equal (in which case eigenvalue has 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.

{{main|Eigenvalue perturbation}}

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

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

:$\backslash mathbf\{B\}\; =\; \backslash mathbf\{P\}\backslash mathbf\{A\}\backslash mathbf\{P\}^\{-1\}$

The matrix P is called a similarity matrix, and is necessarily invertible.

{{main|Unitary matrix}}

{{other uses|Canonical form}}

{{main|Row echelon form}}

{{main|Jordan normal form}}

{{main|Weyr canonical form}}

{{main|Frobenius normal form}}

{{main|LU decomposition}}

LU decomposition splits a matrix into a matrix product of an upper triangular matrix and a lower triangle matrix.

{{Main|Matrix norm}}

Since matrices form vector spaces, one can form axioms (analogous to those of vectors) to define a "size" of a particular matrix. The norm of a matrix is a positive real number.

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:Some authors, e.g. Horn and Johnson, use triple vertical bars instead of double: |||A|||.

• Nonnegative:

::$\backslash |\; \backslash mathbf\{A\}\; \backslash |\; \backslash ge\; 0$

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

• Scalar multiplication:

::$\backslash |\backslash alpha\; \backslash mathbf\{A\}\backslash |=|\backslash alpha|\; \backslash |\backslash mathbf\{A\}\backslash |$

• The triangular inequality:

::$\backslash |\backslash mathbf\{A\}+\backslash mathbf\{B\}\backslash |\; \backslash leq\; \backslash |\backslash mathbf\{A\}\backslash |+\backslash |\backslash mathbf\{B\}\backslash |$

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:

:$\backslash |\backslash mathbf\{A\}\backslash |\; =\; \backslash sqrt\{\backslash mathbf\{A\}:\backslash mathbf\{A\}\}\; =\; \backslash sqrt\{\backslash sum\_\{i=1\}^m\; \backslash sum\_\{j=1\}^n\; (A\_\{ij\})^2\}$

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

{{main|Positive definite matrix}}

{{main|Function (mathematics)}}

Matrix elements are not restricted to constant numbers, they can be mathematical variables.

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

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

• Mathematical analysis
• Tensor analysis
• Matrix calculus
• Numerical analysis

• Tensor product
• Spectrum of an operator
• Matrix geometrical series

• Orthogonal matrix, unitary matrix
• Symmetric matrix, antisymmetric matrix
• Stochastic matrix

• Matrix polynomial
• Matrix exponential

{{Reflist|group="note"|1}}

{{reflist}}

• {{cite book|title=Matrix Analysis and Applied Linear Algebra Book and Solutions Manual|author=C. Meyer|year=2000 |publisher=SIAM|isbn=089-871-454-0|volume=2|url=https://books.google.com/books?id=Zg4M0iFlbGcC&q=Matrix+Analysis}}
• {{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=978-038-733-195-9|series=Undergraduate Texts in Mathematics|url=https://books.google.com/books?id=8qwTb9P-iW8C&q=Matrix+Analysis}}
• {{cite book|title=Matrix Analysis|author=Rajendra Bhatia|year=1997|volume=169|series=Matrix Analysis Series|publisher=Springer|isbn=038-794-846-5|url=https://books.google.com/books?id=F4hRy1F1M6QC&q=matrix+analysis}}
• {{cite book|title=Computational Matrix Analysis|author=Alan J. Laub|year=2012|publisher=SIAM|isbn=978-161-197-221-4|url=https://books.google.com/books?id=RJBZBuHpVjEC&q=Matrix+Analysis}}

Category:Linear algebra

Category:Matrices (mathematics)

Category:Numerical analysis