Matrix
A rectangular array
(1) 
consisting of rows and columns, the entries of which belong to some set . (1) is called also an dimensional matrix over , or a matrix of dimensions over . Let denote the set of all dimensional matrices over . If , then (1) is called a square matrix of order . The set of all square matrices of order over is denoted by .
Alternative notations for matrices are:
In the most important cases the role of is played by the field of real numbers, the field of complex numbers, an arbitrary field, a ring of polynomials, the ring of integers, a ring of functions, or an arbitrary associative ring. The operations of addition and multiplication defined on are carried over naturally to matrices over , and in this way one is led to the matrix calculus — the subject matter of the theory of matrices.
The notion of a matrix arose first in the middle of the 19th century in the investigations of W. Hamilton, and A. Cayley. Fundamental results in the theory of matrices are due to K. Weierstrass, C. Jordan and G. Frobenius. I.A. LappoDanilevskii has developed the theory of analytic functions of several matrix variables and has applied it to the study of systems of linear differential equations.
Contents
Operations with matrices.
Let be an associative ring and let . Then the sum of the matrices and is, by definition,
Clearly, and addition of matrices is associative and commutative. The null matrix in is the matrix 0, all entries of which are zero. For every ,
Let and . The product of the two matrices and is defined by the rule
where
The product of two elements of is always defined and belongs to . Multiplication of matrices is associative: If , and , then
and . The distributivity rule also holds: For and ,
(2) 
In particular, (2) holds also for . Consequently, is an associative ring. If is a ring with an identity, then the matrix
is the identity of the ring :
for all . Multiplication of matrices is not commutative: If , for every associative ring with an identity there are matrices such that .
Let , ; the product of the matrix by the element (number, scalar) is, by definition, the matrix . Then
Let be a ring with an identity. The matrix is defined as the element of the only nonzero entry of which is the entry , which equals 1, , . For every ,
If is a field, then is an dimensional vector space over , and the matrices form a basis in this space.
Block matrices.
Let , , where and are positive integers. Then a matrix can be written in the form
(3) 
where , , . The matrix (3) is called a block matrix. If , , , and is written in the form
then
For example, if , then may be regarded as , where .
The matrix of the form
where and is the null matrix, is denoted by and is called block diagonal. The following holds:
provided that the orders of and coincide for .
Square matrices over a field.
Let be a field, let and let be the determinant of the matrix . is said to be nondegenerate (or nonsingular) if . A matrix is called the inverse of if . The invertibility of in is equivalent to its nondegeneracy, and
where is the cofactor of the entry , . For ,
The set of all invertible elements of is a group under multiplication, called the general linear group and denoted by . The powers of a matrix are defined as follows
and if is invertible, then . For the polynomial
the matrix polynomial
is defined.
Every matrix from gives rise to a linear transformation of the dimensional vector space over . Let be a basis in and let be a linear transformation of . Then is uniquely determined by the set of vectors
Moreover,
(4) 
where . The matrix is called the matrix of the transformation in the basis . For a fixed basis, the matrix is the matrix of the linear transformation , while is the matrix of if is the matrix of the linear transformation . Equality (4) may be written in the form
Suppose that is a second basis in . Then , , and is the matrix of the transformation in the basis . Two matrices are similar if there is a matrix such that . Here, also, and the ranks of the matrices and coincide. The linear transformation is called nondegenerate, or nonsingular, if ; is nondegenerate if and only if its matrix is nondegenerate. If is regarded as the space of columns , then every linear transformation in is given by left multiplication of the columns by some : , and the matrix of in the basis
coincides with . A matrix is singular (or degenerate) if and only if there is a column , , such that .
Transposition and matrices of special form.
Let . Then the matrix , where , is called the transpose of . Alternative notations are and . Let . Then , where is the complex conjugate of the number , is called the complex conjugate of . The matrix , where , is called the Hermitian conjugate of . Many matrices used in applications are given special names:'
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Polynomial matrices.
Let be a field and let be the ring of all polynomials in the variable with coefficients from . A matrix over is called a polynomial matrix. For the elements of the ring one introduces the following elementary operations: 1) multiplication of a row or column of a matrix by a nonzero element of the field ; and 2) addition to a row (column) of another row (respectively, column) of the given matrix, multiplied by a polynomial from . Two matrices are called equivalent if can be obtained from through a finite number of elementary operations.
Let
where a) ; b) is divisible by for ; and c) the coefficient of the leading term in is equal to 1. Then is called a canonical polynomial matrix. Every equivalence class of elements of the ring contains a unique canonical matrix. If , where
is a canonical matrix, then the polynomials
are called the invariant factors of ; the number is identical with the rank of . A matrix has an inverse in if and only if . The last condition is in turn equivalent to . Two matrices are equivalent if and only if
where , .
Let . The matrix
is called the characteristic matrix of and is called the characteristic polynomial of . For every polynomial of the form
there is an such that
Such is, for example, the matrix
(*) 
The characteristic polynomials of two similar matrices coincide. However, the fact that two matrices have identical characteristic polynomials does not necessarily entail the fact that the matrices are similar. A similarity criterion is: Two matrices are similar if and only if the polynomial matrices and are equivalent. The set of all matrices from having a given characteristic polynomial is partitioned into a finite number of classes of similar matrices; this set reduces to a single class if and only if does not have multiple factors in .
Let , , , and suppose that , where . Then is called an eigen vector of and is called an eigen value of . An element is an eigen value of a matrix if and only if it is a root of the characteristic polynomial of . The set of all columns such that for a fixed eigen value of is a subspace of . The dimension of this subspace equals the defect (or deficiency) of the matrix (, where is the rank of ). The number does not exceed the multiplicity of the root , but need not coincide with it. A matrix is similar to a diagonal matrix if and only if it has linearly independent eigen vectors. If for an ,
and the roots are distinct, then the following holds: is similar to a diagonal matrix if and only if for each , , the defect of coincides with . In particular, every matrix with distinct eigen values is similar to a diagonal matrix. Over an algebraically closed field every matrix from is similar to some triangular matrix from . The Hamilton–Cayley theorem: If is the characteristic polynomial of a matrix , then is the null matrix.
By definition, the minimum polynomial of a matrix is the polynomial with the properties: ) ; ) the coefficient of the leading term equals 1; and ) if and the degree of is smaller than the degree of , then . Every matrix has a unique minimum polynomial. If and , then the minimum polynomial of divides . The minimum polynomial and the characteristic polynomial of coincide with the last invariant factor, and, respectively, the product of all invariant factors, of the matrix . The minimum polynomial of equals
where is the greatest common divisor of the minors (cf. Minor) of order of the matrix . A matrix is similar to a diagonal matrix over the field if and only if its minimum polynomial is a product of distinct linear factors in the ring .
A matrix is called nilpotent if for some integer . A matrix is nilpotent if and only if . Every nilpotent matrix from is similar to some triangular matrix with zeros on the diagonal.
References
[1]  V.V. Voevodin, "Algèbre linéare" , MIR (1976) (Translated from Russian) 
[2]  F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian) 
[3]  A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian) 
[4]  A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) 
[5]  A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian) 
[6]  I.B. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra" , Pergamon (1965) (Translated from Russian) 
[7]  A.R.I. Tyshkevich, "Linear algebra and analytic geometry" , Minsk (1976) (In Russian) 
[8]  R. Bellman, "Introduction to matrix analysis" , McGrawHill (1970) 
[9]  N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , AddisonWesley (1974) pp. Chapt.1;2 (Translated from French) 
[10]  P. Lancaster, "Theory of matrices" , Acad. Press (1969) 
[11]  M. Marcus, H. Minc, "A survey of matrix theory and matrix inequalities" , Allyn & Bacon (1964) 
Comments
The result on canonical polynomial matrices quoted above has a natural generalization to matrices over principal ideal domains. An matrix over a principal ideal domain of the form
(a1) 
with divisible by , , is said to be in Smith canonical form. Every matrix over a principal ideal domain is equivalent to one in Smith canonical form in the sense that there are an matrix and an matrix such that and are invertible in and , respectively, and such that is in Smith canonical form.
A matrix of the form (a1) is said to be in companion form, especially in linear systems and control theory where the theory of (polynomial) matrices finds many applications.
References
[a1]  P.M. Cohn, "Algebra" , 1 , Wiley (1974) pp. Sect. 10.6 
[a2]  W.A. Wolovich, "Linear multivariable systems" , Springer (1974) 
[a3]  R.E. Kalman, P.L. Falb, M.A. Arbib, "Topics in mathematical systems theory" , PrenticeHall (1969) 
Matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix&oldid=17958