Number
int64 1
7.61k
| Text
stringlengths 2
3.11k
|
|---|---|
3,901
|
Java's primitive data type byte is defined as eight bits. It is a signed data type, holding values from −128 to 127.
|
3,902
|
.NET programming languages, such as C#, define byte as an unsigned type, and the sbyte as a signed data type, holding values from 0 to 255, and −128 to 127, respectively.
|
3,903
|
In data transmission systems, the byte is used as a contiguous sequence of bits in a serial data stream, representing the smallest distinguished unit of data. For asynchronous communication a full transmission unit usually additionally includes a start bit, 1 or 2 stop bits, and possibly a parity bit, and thus its size may vary from seven to twelve bits for five to eight bits of actual data. For synchronous communication the error checking usually uses bytes at the end of a frame.
|
3,904
|
For example,
|
3,905
|
Matrices are used to represent linear maps and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents the composition of linear maps.
|
3,906
|
Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such.
|
3,907
|
Square matrices, matrices with the same number of rows and columns, play a major role in matrix theory. Square matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring. The determinant of a square matrix is a number associated to the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has a nonzero determinant, and the eigenvalues of a square matrix are the roots of a polynomial determinant.
|
3,908
|
In geometry, matrices are widely used for specifying and representing geometric transformations and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimension. Matrices are used in most areas of mathematics and most scientific fields, either directly, or through their use in geometry and numerical analysis.
|
3,909
|
Matrix theory is the branch of mathematics that focuses on the study of matrices. It was initially a sub-branch of linear algebra, but soon grew to include subjects related to graph theory, algebra, combinatorics and statistics.
|
3,910
|
A matrix is a rectangular array of numbers , called the entries of the matrix. Matrices are subject to standard operations such as addition and multiplication. Most commonly, a matrix over a field F is a rectangular array of elements of F. A real matrix and a complex matrix are matrices whose entries are respectively real numbers or complex numbers. More general types of entries are discussed below. For instance, this is a real matrix:
|
3,911
|
The numbers, symbols, or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.
|
3,912
|
Matrices with a single row are called row vectors, and those with a single column are called column vectors. A matrix with the same number of rows and columns is called a square matrix. A matrix with an infinite number of rows or columns is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.
|
3,913
|
There are a number of basic operations that can be applied on matrices. Some, such as transposition and submatrix do not depend on the nature of the entries. Others, such as matrix addition, scalar multiplication, matrix multiplication, and row operations involve operations on matrix entries and therefore require that matrix entries are numbers or belong to a field or a ring.
|
3,914
|
In this section, it is supposed that matrix entries belong to a fixed ring, that is typically a field of numbers.
|
3,915
|
The sum A+B of two m-by-n matrices A and B is calculated entrywise:
|
3,916
|
For example,
|
3,917
|
The product cA of a number c and a matrix A is computed by multiplying every entry of A by c:
|
3,918
|
This operation is called scalar multiplication, but its result is not named "scalar product" to avoid confusion, since "scalar product" is often used as a synonym for "inner product". For example:
|
3,919
|
The subtraction of two m×n matrices is defined by composing matrix addition with scalar multiplication by –1:
|
3,920
|
The transpose of an m-by-n matrix A is the n-by-m matrix AT formed by turning rows into columns and vice versa:
|
3,921
|
For example:
|
3,922
|
Familiar properties of numbers extend to these operations on matrices: for example, addition is commutative, that is, the matrix sum does not depend on the order of the summands: A + B = B + A.
The transpose is compatible with addition and scalar multiplication, as expressed by T = c and T = AT + BT. Finally, T = A.
|
3,923
|
Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B:
|
3,924
|
where 1 ≤ i ≤ m and 1 ≤ j ≤ p. For example, the underlined entry 2340 in the product is calculated as + + = 2340:
|
3,925
|
Matrix multiplication satisfies the rules C = A , and C = AC + BC as well as C = CA + CB , whenever the size of the matrices is such that the various products are defined. The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and m ≠ k. Even if both products are defined, they generally need not be equal, that is:
|
3,926
|
In other words, matrix multiplication is not commutative, in marked contrast to numbers, whose product is independent of the order of the factors. An example of two matrices not commuting with each other is:
|
3,927
|
Besides the ordinary matrix multiplication just described, other less frequently used operations on matrices that can be considered forms of multiplication also exist, such as the Hadamard product and the Kronecker product. They arise in solving matrix equations such as the Sylvester equation.
|
3,928
|
There are three types of row operations:
|
3,929
|
row addition, that is adding a row to another.
|
3,930
|
row multiplication, that is multiplying all entries of a row by a non-zero constant;
|
3,931
|
row switching, that is interchanging two rows of a matrix;
|
3,932
|
These operations are used in several ways, including solving linear equations and finding matrix inverses.
|
3,933
|
A submatrix of a matrix is a matrix obtained by deleting any collection of rows and/or columns. For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2:
|
3,934
|
The minors and cofactors of a matrix are found by computing the determinant of certain submatrices.
|
3,935
|
A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain. Other authors define a principal submatrix as one in which the first k rows and columns, for some number k, are the ones that remain; this type of submatrix has also been called a leading principal submatrix.
|
3,936
|
Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an m-by-n matrix, x designates a column vector of n variables x1, x2, ..., xn, and b is an m×1-column vector, then the matrix equation
|
3,937
|
is equivalent to the system of linear equations
|
3,938
|
Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If n = m and the equations are independent, then this can be done by writing
|
3,939
|
where A−1 is the inverse matrix of A. If A has no inverse, solutions—if any—can be found using its generalized inverse.
|
3,940
|
Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real m-by-n matrix A gives rise to a linear transformation Rn → Rm mapping each vector x in Rn to the product Ax, which is a vector in Rm. Conversely, each linear transformation f: Rn → Rm arises from a unique m-by-n matrix A: explicitly, the -entry of A is the ith coordinate of f, where ej = is the unit vector with 1 in the jth position and 0 elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f.
|
3,941
|
For example, the 2×2 matrix
|
3,942
|
The following table shows several 2×2 real matrices with the associated linear maps of R2. The blue original is mapped to the green grid and shapes. The origin is marked with a black point.
|
3,943
|
Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps: if a k-by-m matrix B represents another linear map g: Rm → Rk, then the composition g ∘ f is represented by BA since
|
3,944
|
The last equality follows from the above-mentioned associativity of matrix multiplication.
|
3,945
|
The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. Equivalently it is the dimension of the image of the linear map represented by A. The rank–nullity theorem states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix.
|
3,946
|
A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied.
The entries aii form the main diagonal of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix.
|
3,947
|
If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. Similarly if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries outside the main diagonal are zero, A is called a diagonal matrix.
|
3,948
|
The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example,
|
3,949
|
It is a square matrix of order n, and also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:
|
3,950
|
A nonzero scalar multiple of an identity matrix is called a scalar matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.
|
3,951
|
A square matrix A that is equal to its transpose, that is, A = AT, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −AT, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A.
|
3,952
|
By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below.
|
3,953
|
A square matrix A is called invertible or non-singular if there exists a matrix B such that
|
3,954
|
where In is the n×n identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the inverse matrix of A, denoted A−1.
|
3,955
|
A symmetric real matrix A is called positive-definite if the associated quadratic form
|
3,956
|
has a positive value for every nonzero vector x in Rn. If f only yields negative values then A is negative-definite; if f does produce both negative and positive values then A is indefinite. If the quadratic form f yields only non-negative values , the symmetric matrix is called positive-semidefinite ; hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.
|
3,957
|
A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible. The table at the right shows two possibilities for 2-by-2 matrices.
|
3,958
|
Allowing as input two different vectors instead yields the bilinear form associated to A:
|
3,959
|
In the case of complex matrices, the same terminology and result apply, with symmetric matrix, quadratic form, bilinear form, and transpose xT replaced respectively by Hermitian matrix, Hermitian form, sesquilinear form, and conjugate transpose xH.
|
3,960
|
An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors . Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse:
|
3,961
|
which entails
|
3,962
|
where In is the identity matrix of size n.
|
3,963
|
An orthogonal matrix A is necessarily invertible , unitary , and normal . The determinant of any orthogonal matrix is either +1 or −1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant +1 is a pure rotation without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant -1 reverses the orientation, i.e., is a composition of a pure reflection and a rotation. The identity matrices have determinant 1, and are pure rotations by an angle zero.
|
3,964
|
The complex analogue of an orthogonal matrix is a unitary matrix.
|
3,965
|
The trace, tr of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned above, the trace of the product of two matrices is independent of the order of the factors:
|
3,966
|
This is immediate from the definition of matrix multiplication:
|
3,967
|
It follows that the trace of the product of more than two matrices is independent of cyclic permutations of the matrices, however this does not in general apply for arbitrary permutations ≠ tr, in general). Also, the trace of a matrix is equal to that of its transpose, that is,
|
3,968
|
The determinant of a square matrix A or |A|) is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area or volume of the image of the unit square , while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
|
3,969
|
The determinant of 2-by-2 matrices is given by
|
3,970
|
The determinant of 3-by-3 matrices involves 6 terms . The more lengthy Leibniz formula generalises these two formulae to all dimensions.
|
3,971
|
The determinant of a product of square matrices equals the product of their determinants:
|
3,972
|
Adding a multiple of any row to another row, or a multiple of any column to another column does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower triangular matrix, and for such matrices, the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, that is, determinants of smaller matrices. This expansion can be used for a recursive definition of determinants , that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.
|
3,973
|
are called an eigenvalue and an eigenvector of A, respectively. The number λ is an eigenvalue of an n×n-matrix A if and only if A−λIn is not invertible, which is equivalent to
|
3,974
|
The polynomial pA in an indeterminate X given by evaluation of the determinant det is called the characteristic polynomial of A. It is a monic polynomial of degree n. Therefore the polynomial equation pA = 0 has at most n different solutions, that is, eigenvalues of the matrix. They may be complex even if the entries of A are real. According to the Cayley–Hamilton theorem, pA = 0, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.
|
3,975
|
Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms or iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a sequence of vectors xn converging to an eigenvector when n tends to infinity.
|
3,976
|
To choose the most appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms. The domain studying these matters is called numerical linear algebra. As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stability.
|
3,977
|
Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, for example, multiplication of matrices. Calculating the matrix product of two n-by-n matrices using the definition given above needs n3 multiplications, since for any of the n2 entries of the product, n multiplications are necessary. The Strassen algorithm outperforms this "naive" algorithm; it needs only n2.807 multiplications. A refined approach also incorporates specific features of the computing devices.
|
3,978
|
In many practical situations additional information about the matrices involved is known. An important case are sparse matrices, that is, matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method.
|
3,979
|
An algorithm is, roughly speaking, numerically stable, if little deviations in the input values do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace expansion denotes the adjugate matrix of A)
|
3,980
|
may lead to significant rounding errors if the determinant of the matrix is very small. The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix's inverse.
|
3,981
|
Most computer programming languages support arrays but are not designed with built-in commands for matrices. Instead, available external libraries provide matrix operations on arrays, in nearly all currently used programming languages. Matrix manipulation was among the earliest numerical applications of computers. The original Dartmouth BASIC had built-in commands for matrix arithmetic on arrays from its second edition implementation in 1964. As early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC commands for matrices. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices. As of 2023, most computers have some form of built-in matrix operations at a low-level implementing the standard BLAS specification, upon which most higher-level matrix and linear algebra libraries rely. While most of these libraries require a professional level of coding, LAPACK can be accessed by higher-level bindings such as NumPy/SciPy, R, GNU Octave, MATLAB.
|
3,982
|
There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix decomposition or matrix factorization techniques. The interest of all these techniques is that they preserve certain properties of the matrices in question, such as determinant, rank, or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.
|
3,983
|
The LU decomposition factors matrices as a product of lower and an upper triangular matrices . Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form. Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row. Singular value decomposition expresses any matrix A as a product UDV∗, where U and V are unitary matrices and D is a diagonal matrix.
|
3,984
|
The eigendecomposition or diagonalization expresses A as a product VDV−1, where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called diagonalizable. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal form, that is to say matrices whose only nonzero entries are the eigenvalues λ1 to λn of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right. Given the eigendecomposition, the nth power of A can be calculated via
|
3,985
|
and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the matrix exponential eA, a need frequently arising in solving linear differential equations, matrix logarithms and square roots of matrices. To avoid numerically ill-conditioned situations, further algorithms such as the Schur decomposition can be employed.
|
3,986
|
Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension is tensors, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realized as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers. Matrices, subject to certain requirements tend to form groups known as matrix groups. Similarly under certain conditions matrices form rings known as matrix rings. Though the product of matrices is not in general commutative yet certain matrices form fields known as matrix fields.
|
3,987
|
This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any field, that is, a set where addition, subtraction, multiplication, and division operations are defined and well-behaved, may be used instead of R or C, for example rational numbers or finite fields. For example, coding theory makes use of matrices over finite fields. Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a larger field than that of the entries of the matrix; for instance, they may be complex in the case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an algebraically closed field, such as C, from the outset.
|
3,988
|
More generally, matrices with entries in a ring R are widely used in mathematics. Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set M ) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn. If the ring R is commutative, that is, its multiplication is commutative, then the ring M is also an associative algebra over R. The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible. Matrices over superrings are called supermatrices.
|
3,989
|
Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any ring; but their sizes must fulfill certain compatibility conditions.
|
3,990
|
Linear maps Rn → Rm are equivalent to m-by-n matrices, as described above. More generally, any linear map f: V → W between finite-dimensional vector spaces can be described by a matrix A = , after choosing bases v1, ..., vn of V, and w1, ..., wm of W , which is such that
|
3,991
|
In other words, column j of A expresses the image of vj in terms of the basis vectors wi of W; thus this relation uniquely determines the entries of the matrix A. The matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices. Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the transpose of the linear map given by A, with respect to the dual bases.
|
3,992
|
More generally, the set of m×n matrices can be used to represent the R-linear maps between the free modules Rm and Rn for an arbitrary ring R with unity. When n = m composition of these maps is possible, and this gives rise to the matrix ring of n×n matrices representing the endomorphism ring of Rn.
|
3,993
|
A group is a mathematical structure consisting of a set of objects together with a binary operation, that is, an operation combining any two objects to a third, subject to certain requirements. A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group. Since a group every element must be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups.
|
3,994
|
Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a subgroup of their general linear group, called a special linear group. Orthogonal matrices, determined by the condition
|
3,995
|
form the orthogonal group. Every orthogonal matrix has determinant 1 or −1. Orthogonal matrices with determinant 1 form a subgroup called special orthogonal group.
|
3,996
|
Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group. General groups can be studied using matrix groups, which are comparatively well understood, by means of representation theory.
|
3,997
|
It is also possible to consider matrices with infinitely many rows and/or columns even though, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry . The basic operations of addition, subtraction, scalar multiplication, and transposition can still be defined without problem; however, matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general.
|
3,998
|
If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix A to describe a linear map f: V→W, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a linear combination of basis vectors, so that written as a vector v of coefficients, only finitely many entries vi are nonzero. Now the columns of A describe the images by f of individual basis vectors of V in the basis of W, which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of A however: in the product A·v there are only finitely many nonzero coefficients of v involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover, this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries because each of those columns does. Products of two matrices of the given type are well defined , are of the same type, and correspond to the composition of linear maps.
|
3,999
|
If R is a normed ring, then the condition of row or column finiteness can be relaxed. With the norm in place, absolutely convergent series can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously, the matrices whose row sums are absolutely convergent series also form a ring.
|
4,000
|
Infinite matrices can also be used to describe operators on Hilbert spaces, where convergence and continuity questions arise, which again results in certain constraints that must be imposed. However, the explicit point of view of matrices tends to obfuscate the matter, and the abstract and more powerful tools of functional analysis can be used instead.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.