Matrix analysis

In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. 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 sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).

Matrix spaces

The set of all m×n matrices over a number field F denoted in this article M_mn(F) form a vector space. Examples of F include the set of integers ℤ, the real numbers ℝ, and set of complex numbers 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):

$\mathbf {A} ,\mathbf {B} \in M_{mn}(F)\,,\quad \mathbf {A} +\mathbf {B} \in M_{mn}(F)$

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

$\alpha \in F\,,\quad \alpha \mathbf {A} \in M_{mn}(F)$

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

$\alpha \mathbf {A} +\beta \mathbf {B} \in M_{mn}(F)$

where α and β are numbers in F.

Determinants

Main article: 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). 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

Main article: Eigenvalues and eigenvectors

Definitions

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

$\mathbf {A} \mathbf {x} =\lambda \mathbf {x}$

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 solutions to the characteristic polynomial:

$\det(\mathbf {A} -\lambda \mathbf {I} )=0$

where I is the n×n identity matrix.

Roots of polynomials can all be different, or some may be equal (in which case the eigenvalue has multiplicity). After solving for the eigenvalues, the eigenvectors corresponding to the eigenvalues can be found by the defining equation.

Perturbations of eigenvalues

Main article: Eigenvalue perturbation

Matrix similarity

Main article: Matrix similarity

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

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

The matrix P is called a similarity matrix.

Unitary similarity

Main article: Unitary matrix

Canonical forms

For other uses, see Canonical form.

Row echelon form

Main article: Row echelon form

Jordan normal form

Main article: Jordan normal form

Weyr canonical form

Main article: Weyr canonical form

Frobenius normal form

Main article: Frobenius normal form

Triangular factorization

LU decomposition

Main article: LU decomposition

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

Matrix norms

Main article: 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.

Definition and axioms

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:

• Positive definite:

$\|\mathbf {A} \|\geq 0$

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

• Scalar multiplication:

$\|\alpha \mathbf {A} \|=|\alpha |\|\mathbf {A} \|$

• The triangular inequality:

$\|\mathbf {A} +\mathbf {B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|$

Frobenius norm

The Frobenius norm is analogous to the dot product of Euclidean vectors; square every matrix element and add up the results, then take the positive square root:

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

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

Positive definite and semidefinite matrices

Main article: Positive definite matrix

Functions

Main article: Function (mathematics)

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

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.

See also

• Mathematical analysis
• Tensor analysis
• Matrix calculus
• Numerical analysis
• Determinant
• Matrix similarity
• Tensor product
• Spectrum of an operator
• Stochastic matrix

References

Further reading

• N. J. Higham (2000). Functions of Matrices: Theory and Computation. SIAM. ISBN 089-871-777-9.

• C. Meyer (2000). Matrix Analysis and Applied Linear Algebra Book and Solutions Manual. Matrix Analysis and Applied Linear Algebra. Vol. 2. SIAM. ISBN 089-871-454-0.

• T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 038-733-195-6.

• Rajendra Bhatia (1997). Matrix Analysis. Matrix Analysis Series. Vol. 169. Springer. ISBN 038-794-846-5.

• Alan J. Laub (2012). Computational Matrix Analysis. SIAM. ISBN 161-197-221-3.