NationMaster - Encyclopedia: Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...

1 Ordinary matrix product
2 Scalar multiplication
3 Hadamard product
4 Kronecker product
5 Common properties
6 Frobenius inner product
7 See also
8 References
9 External links

Ordinary matrix product

This is the most often used and most important way to multiply matrices. It is defined between two matrices only if the number of columns of the first matrix is the same as the number of rows of the second matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their product is an m-by-p matrix denoted by AB (or sometimes A · B). If C = AB, and c_i,j denotes the entry in C at position (i,j), then

$c_{i,j} = sum_{r=1}^n a_{i,r}b_{r,j} = a_{i,1}b_{1,j} + a_{i,2}b_{2,j} + cdots + a_{i,n}b_{n,j}.$

for each pair i and j with 1 ≤ i ≤ m and 1 ≤ j ≤ p. The algebraic system of "matrix units" summarises the abstract properties of this kind of multiplication. In mathematics, a matrix unit is an idealisation of the concept of a matrix, with a focus on the algebraic properties of matrix multiplication. ...

Calculating directly from the definition

The picture to the left shows how to calculate the (1,2) element and the (3,3) element of AB if A is a 4×2 matrix, and B is a 2×3 matrix. Elements from each matrix are paired off in the direction of the arrows; each pair is multiplied and the products are added. The location of the resulting number in AB corresponds to the row and column that were considered.

$(mathbf{AB})_{1,2} = sum_{r=1}^2 a_{1,r}b_{r,2} = a_{1,1}b_{1,2}+a_{1,2}b_{2,2}$

$(mathbf{AB})_{3,3} = sum_{r=1}^2 a_{3,r}b_{r,3} = a_{3,1}b_{1,3}+a_{3,2}b_{2,3}$

For example:

$begin{bmatrix} 1 & 0 & 2 -1 & 3 & 1 end{bmatrix} cdot begin{bmatrix} 3 & 1 2 & 1 1 & 0 end{bmatrix} = begin{bmatrix} 1 times 3 + 0 times 2 + 2 times 1 & 1 times 1 + 0 times 1 + 2 times 0 -1 times 3 + 3 times 2 + 1 times 1 & -1 times 1 + 3 times 1 + 1 times 0 end{bmatrix} = begin{bmatrix} 5 & 1 4 & 2 end{bmatrix}$

The coefficients-vectors method

This matrix multiplication can also be considered from a slightly different viewpoint: it adds vectors together after being multiplied by different coefficients. If A and B are matrices given by: This article is about vectors that have a particular relation to the spatial coordinates. ... In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ...

$mathbf{A} = begin{bmatrix} a_{1,1} & a_{1,2} & dots a_{2,1} & a_{2,2} & dots vdots & vdots & ddots end{bmatrix}$ and $mathbf{B} = begin{bmatrix} b_{1,1} & b_{1,2} & dots b_{2,1} & b_{2,2} & dots vdots & vdots & ddots end{bmatrix} = begin{bmatrix} B_1 B_2 vdots end{bmatrix}$

then

$mathbf{AB} = begin{bmatrix} a_{1,1} B_1 + a_{1,2} B_2 + cdots a_{2,1} B_1 + a_{2,2} B_2 + cdots vdots end{bmatrix}$

The example revisited:

$begin{bmatrix} 1 & 0 & 2 -1 & 3 & 1 end{bmatrix} cdot begin{bmatrix} 3 & 1 2 & 1 1 & 0 end{bmatrix} = begin{bmatrix} 1 begin{bmatrix} 3 & 1 end{bmatrix} + 0 begin{bmatrix} 2 & 1 end{bmatrix} + 2 begin{bmatrix} 1 & 0 end{bmatrix} -1 begin{bmatrix} 3 & 1 end{bmatrix} + 3 begin{bmatrix} 2 & 1 end{bmatrix} + 1 begin{bmatrix} 1 & 0 end{bmatrix} end{bmatrix} = begin{bmatrix} begin{bmatrix} 3 & 1 end{bmatrix} + begin{bmatrix} 0 & 0 end{bmatrix} + begin{bmatrix} 2 & 0 end{bmatrix} begin{bmatrix} -3 & -1 end{bmatrix} + begin{bmatrix} 6 & 3 end{bmatrix} + begin{bmatrix} 1 & 0 end{bmatrix} end{bmatrix}$

$= begin{bmatrix} 5 & 1 4 & 2 end{bmatrix}$

The rows in the matrix on the left are the list of coefficients. The matrix on the right is the list of vectors. In the example, the first row is [1 0 2], and thus we take 1 times the first vector, 0 times the second vector, and 2 times the third vector.

The equation can be simplified further by using outer products: Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ...

$mathbf{A} = begin{bmatrix} A_1 & A_2 & dots end{bmatrix} implies mathbf{AB} = sum_i A_iB_i$

The terms of this sum are matrices of the same shape, each describing the effect of one column of A and one row of B on the result. The columns of A can be seen as a coordinate system of the transform, i.e. given a vector x we have $mathbf{A}x=A_1x_1+A_2x_2+cdots$ where $x i$ are coordinates along the $A i$ "axes". The terms $A i B i$ are like $A i x i$ , except that $B i$ contains the ith coordinate for each column vector of B, each of which is transformed independently in parallel.

The example revisited:

$begin{bmatrix} 1 & 0 & 2 -1 & 3 & 1 end{bmatrix} cdot begin{bmatrix} 3 & 1 2 & 1 1 & 0 end{bmatrix} = begin{bmatrix}1 -1end{bmatrix}begin{bmatrix}3 & 1end{bmatrix}+ begin{bmatrix}0 3end{bmatrix}begin{bmatrix}2 & 1end{bmatrix}+ begin{bmatrix}2 1end{bmatrix}begin{bmatrix}1 & 0end{bmatrix}$

$= begin{bmatrix} 1 cdot 3 & 1 cdot 1 -1 cdot 3 & -1 cdot 1 end{bmatrix}+ begin{bmatrix} 0 cdot 2 & 0 cdot 1 3 cdot 2 & 3 cdot 1 end{bmatrix}+ begin{bmatrix} 2 cdot 1 & 2 cdot 0 1 cdot 1 & 1 cdot 0 end{bmatrix} = begin{bmatrix} 5 & 1 4 & 2 end{bmatrix}$

The vectors $begin{bmatrix}3 & 2 & 1end{bmatrix}^top$ and $begin{bmatrix}1 & 1 & 0end{bmatrix}^top$ have been transformed to $begin{bmatrix}5 & 4end{bmatrix}^top$ and $begin{bmatrix}1 & 2end{bmatrix}^top$ in parallel. One could also transform them one by one with the same steps:

$begin{bmatrix} 1 & 0 & 2 -1 & 3 & 1 end{bmatrix} cdot begin{bmatrix} 3 2 1 end{bmatrix} = begin{bmatrix}1 -1end{bmatrix}3+ begin{bmatrix}0 3end{bmatrix}2+ begin{bmatrix}2 1end{bmatrix}1 = begin{bmatrix} 1cdot 3 -1cdot 3end{bmatrix}+ begin{bmatrix} 0cdot 2 3cdot 2end{bmatrix}+ begin{bmatrix} 2cdot 1 1cdot 1end{bmatrix} = begin{bmatrix} 5 4 end{bmatrix}$

Vector-lists method

The ordinary matrix product can be thought of as a dot product of a column-list of vectors and a row-list of vectors. If A and B are matrices given by: In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In linear algebra, a column vector is an m Ã— 1 matrix, i. ... In linear algebra, a row vector is a 1 Ã— n matrix, that is, a matrix consisting of a single row: The transpose of a row vector is a column vector. ...

$mathbf{A} = begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} & dots a_{2,1} & a_{2,2} & a_{2,3} & dots a_{3,1} & a_{3,2} & a_{3,3} & dots vdots & vdots & vdots & ddots end{bmatrix} = begin{bmatrix} A_1 A_2 A_3 vdots end{bmatrix}$ and $mathbf{B} = begin{bmatrix} b_{1,1} & b_{1,2} & b_{1,3} & dots b_{2,1} & b_{2,2} & b_{2,3} & dots b_{3,1} & b_{3,2} & b_{3,3} & dots vdots & vdots & vdots & ddots end{bmatrix} = begin{bmatrix} B_1 & B_2 & B_3 & dots end{bmatrix}$

where

A₁ is the row vector of all elements of the form a_1,x A₂ is the row vector of all elements of the form a_2,x etc,

and B₁ is the column vector of all elements of the form b_x,1 B₂ is the column vector of all elements of the form b_x,2 etc,

then

$mathbf{AB} = begin{bmatrix} A_1 A_2 A_3 vdots end{bmatrix} * begin{bmatrix} B_1 & B_2 & B_3 & dots end{bmatrix} = begin{bmatrix} (A_1 cdot B_1) & (A_1 cdot B_2) & (A_1 cdot B_3) & dots (A_2 cdot B_1) & (A_2 cdot B_2) & (A_2 cdot B_3) & dots (A_3 cdot B_1) & (A_3 cdot B_2) & (A_3 cdot B_3) & dots vdots & vdots & vdots & ddots end{bmatrix}.$

Properties

Matrix multiplication is not commutative (that is, AB ≠ BA), except in special cases. It is easy to see why: you cannot expect to switch the proportions with the vectors and get the same result. It is also easy to see how the order of the factors determines the result when one knows that the number of columns in the proportions matrix has to be the same as the number of rows in the vectors matrix: they have to represent the same number of vectors. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

Although matrix multiplication is not commutative, the determinants of AB and BA are always equal (if A and B are square matrices of the same size). See the article on determinants for an explanation. However matrix multiplication is commutative when both matrices are diagonal and of the same dimension.^[1] In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...

This notion of multiplication is important because if A and B are interpreted as linear transformations (which is almost universally done), then the matrix product AB corresponds to the composition of the two linear transformations, with B being applied first. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

Additionally, all notions of matrix multiplication described here share a set of common properties described below. In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. ...

Algorithms

The complexity of matrix multiplication, if carried out naively, is O(n³), but more efficient algorithms do exist. Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication", is based on a clever way of multiplying two 2 × 2 matrices which requires only 7 multiplications (instead of the usual 8). Applying this trick recursively gives an algorithm with a cost of $O( n^{log_{2}7}) approx O(n^{2.807})$ . In practice, though, it is rarely used since it is awkward to implement and it lacks numerical stability. As a branch of the theory of computation in computer science, computational complexity theory investigates the problems related to the amounts of resources required for the execution of algorithms (e. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... In the mathematical discipline of linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm used for matrix multiplication. ... Volker Strassen is a German mathematician. ... Also: 1969 (number) 1969 (movie) 1969 (Stargate SG-1) episode. ... In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ...

The algorithm with the lowest known exponent, which was presented by Don Coppersmith and Shmuel Winograd in 1990, has an asymptotic complexity of O(n^2.376). It is similar to Strassen's algorithm: a clever way is devised for multiplying two k × k matrices with fewer than k³ multiplications, and this technique is applied recursively. It improves on the constant factor in Strassen's algorithm, reducing it to 4.537. However, the constant term implied in the O(n^2.376) result is so large that the Coppersmith–Winograd algorithm is only worthwhile for matrices that are too big to handle on present-day computers (Knuth, 1998). In the mathematical discipline of linear algebra, the Coppersmithâ€“Winograd algorithm is the fastest currently known algorithm for square matrix multiplication. ... Don Coppersmith is a cryptographer and mathematician. ... Shmuel Winograd is a computer scientist, noted for his work on fast algorithms for arithmetic, and in particular for the algorithm known as the Coppersmith-Winograd algorithm. ... This article is about the year. ...

Since any algorithm for multiplying two n × n matrices has to process all 2 × n² entries, there is an asymptotic lower bound of Ω(n²) operations. Raz (2002) proves a lower bound of $Ω(m 2 log m)$ for bounded coefficient arithmetic circuits over the real or complex numbers.

Cohn et al. (2003, 2005) put methods such as the Strassen and Coppersmith–Winograd algorithms in an entirely different, group-theoretic context. They show that if families of wreath products of Abelian with symmetric groups satisfying certain conditions exists, matrix multiplication algorithms with essential quadratic complexity exist. Most researchers believe that this is indeed the case (Robinson, 2005). Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. ...

Scalar multiplication

The scalar multiplication of a matrix A = (a_ij) and a scalar r gives a product r A of the same size as A. The entries of r A are given by

$(rmathbf{A})_{ij} = r cdot a_{ij}. ,$

For example, if

$mathbf{A}=begin{bmatrix} 1 & 2 3 & 4 end{bmatrix}$

then

$7 mathbf{A}=begin{bmatrix} 7 cdot 1 & 7 cdot 2 7 cdot 3 & 7 cdot 4 end{bmatrix} = begin{bmatrix} 7 & 14 21 & 28 end{bmatrix}.$

If we are concerned with matrices over a ring, then the above multiplication is sometimes called the left multiplication while the right multiplication is defined to be In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

$(mathbf{A}r)_{ij} = a_{ij} cdot r. ,$

When the underlying ring is commutative, for example, the real or complex number field, the two multiplications are the same. However, if the ring is not commutative, such as the quaternions, they may be different. For example In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... Graphical representation of quaternion units product as 90Â°-rotation in 4D-space, ij = k, ji = -k, ij = -ji This page describes the mathematical entity. ...

$ibegin{bmatrix} i & 0 0 & j end{bmatrix} = begin{bmatrix} -1 & 0 0 & k end{bmatrix} ne begin{bmatrix} -1 & 0 0 & -k end{bmatrix} = begin{bmatrix} i & 0 0 & j end{bmatrix}i.$

Hadamard product

For two matrices of the same dimensions, we have the Hadamard product, also known as the entrywise product and the Schur product. It can be generalized to hold not only for matrices but also for operators. The Hadamard product of two m-by-n matrices A and B, denoted by A • B, is an m-by-n matrix given by (A•B)_ij = a_ij b_ij. For instance This page is a candidate for speedy deletion. ...

$begin{bmatrix} 1 & 2 3 & 1 end{bmatrix} bullet begin{bmatrix} 0 & 3 2 & 1 end{bmatrix} = begin{bmatrix} 1cdot 0 & 2cdot 3 3cdot 2 & 1cdot 1 end{bmatrix} = begin{bmatrix} 0 & 6 6 & 1 end{bmatrix}$ .

Note that the Hadamard product is a submatrix of the Kronecker product (see below). A submatrix is a matrix formed by taking certain rows and columns from a bigger matrix. ...

The Hadamard product is commutative. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

The Hadamard product is studied by matrix theorists, and it appears in lossy compression algorithms such as JPEG, but it is virtually untouched by linear algebraists. It is discussed in (Horn & Johnson, 1994, Ch. 5). Original Image (lossless PNG, 60. ... JPG redirects here. ...

Kronecker product

Main article: Kronecker product. In mathematics, the Kronecker product, denoted by , is an operation on two matrices of arbitrary size resulting in a block matrix. ...

For any two arbitrary matrices A and B, we have the direct product or Kronecker product A ⊗ B defined as Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ...

$begin{bmatrix} a_{11}B & a_{12}B & cdots & a_{1n}B vdots & vdots & ddots & vdots a_{m1}B & a_{m2}B & cdots & a_{mn}B end{bmatrix}.$

Note that if A is m-by-n and B is p-by-r then A ⊗ B is an mp-by-nr matrix. Again this multiplication is not commutative.

For example

$begin{bmatrix} 1 & 2 3 & 1 end{bmatrix} otimes begin{bmatrix} 0 & 3 2 & 1 end{bmatrix} = begin{bmatrix} 1cdot 0 & 1cdot 3 & 2cdot 0 & 2cdot 3 1cdot 2 & 1cdot 1 & 2cdot 2 & 2cdot 1 3cdot 0 & 3cdot 3 & 1cdot 0 & 1cdot 3 3cdot 2 & 3cdot 1 & 1cdot 2 & 1cdot 1 end{bmatrix} = begin{bmatrix} 0 & 3 & 0 & 6 2 & 1 & 4 & 2 0 & 9 & 0 & 3 6 & 3 & 2 & 1 end{bmatrix}$ .

If A and B represent linear transformations V₁ → W₁ and V₂ → W₂, respectively, then A ⊗ B represents the tensor product of the two maps, V₁ ⊗ V₂ → W₁ ⊗ W₂. In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...

Common properties

Wikibooks The Book of Mathematical Proofs has a page on the topic of

Proofs of properties of matrices

All three notions of matrix multiplication are associative: Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... In mathematics, associativity is a property that a binary operation can have. ...

$mathbf{A} ( mathbf{B C} ) = ( mathbf{A B} ) mathbf{C}$

and distributive: In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

$mathbf{A} ( mathbf{B} + mathbf{C} ) = mathbf{A B} + mathbf{AC}$

and

$( mathbf{A} + mathbf{B} ) mathbf{C} = mathbf{A C} + mathbf{B C}$ .

and compatible with scalar multiplication:

$c ( mathbf{A B} ) = ( c mathbf{A} ) mathbf{B}$

$( mathbf{A} c ) mathbf{B} = mathbf{A} ( c mathbf{B} )$

$( mathbf{A B} ) c = mathbf{A} ( mathbf{B} c )$

Note that these three separate couples of expressions will be equal to each other only if the multiplication and addition on the scalar field are commutative, i.e. the scalar field is a commutative ring. See Scalar multiplication above for a counter-example such as the scalar field of quaternions.

Frobenius inner product

The Frobenius inner product, sometimes denoted A:B is the component-wise inner product of two matrices as though they are vectors. In other words, it is the sum of the entries of the Hadamard product, that is,

$mathbf{A}:mathbf{B}=sum_isum_j A_{ij} B_{ij} = operatorname{trace}(mathbf{A}^T mathbf{B}) = operatorname{trace}(mathbf{A} mathbf{B}^T).$

This inner product induces the Frobenius norm. In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ...

References

Henry Cohn, Robert Kleinberg, Balazs Szegedy, and Chris Umans. Group-theoretic Algorithms for Matrix Multiplication. arXiv:math.GR/0511460. Proceedings of the 46th Annual Symposium on Foundations of Computer Science, 23-25 October 2005, Pittsburgh, PA, IEEE Computer Society, pp. 379–388.
Henry Cohn, Chris Umans. A Group-theoretic Approach to Fast Matrix Multiplication. arXiv:math.GR/0307321. Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 11-14 October 2003, Cambridge, MA, IEEE Computer Society, pp. 438–449.
Coppersmith, D., Winograd S., Matrix multiplication via arithmetic progressions, J. Symbolic Comput. 9, p. 251-280, 1990.
Horn, Roger; Johnson, Charles: "Topics in Matrix Analysis", Cambridge, 1994.
R. Raz. On the complexity of matrix product. In Proceedings of the thirty-fourth annual ACM symposium on Theory of computing. ACM Press, 2002.
Robinson, Sara, Toward an Optimal Algorithm for Matrix Multiplication, SIAM News 38(9), November 2005. PDF
Strassen, Volker, Gaussian Elimination is not Optimal, Numer. Math. 13, p. 354-356, 1969.
Knuth, D.E., The Art of Computer Programming Volume 2: Seminumerical Algorithms. Third Edition, 1998. pp 501.

^ Matrix Multiplication

arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ...

External links

The Simultaneous Triple Product Property and Group-theoretic Results for the Exponent of Matrix Multiplication
WIMS Online Matrix Multiplier
Matrix Multiplication Problems
Matrix Multiplication in C

Categories: Matrix theory | Binary operations

Results from FactBites:

Matrix (mathematics) - Wikipedia, the free encyclopedia (1664 words)
	The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or (i,j)-th entry of A.
	A 1 × n matrix (one row and n columns) is called a row vector, and an m × 1 matrix (one column and m rows) is called a column vector.
	The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.

Matrix multiplication - Wikipedia, the free encyclopedia (1036 words)
	Matrix multiplication is not commutative (that is, AB ≠ BA), except in special cases.
	Although matrix multiplication is not commutative, the determinants of AB and BA are always equal (if A and B are square matrices of the same size).
	Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication", is based on a clever way of multiplying two 2 × 2 matrices which requires only 7 multiplications (instead of the usual 8).

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Contents

Ordinary matrix product GA_googleFillSlot("encyclopedia_square");

Calculating directly from the definition

The coefficients-vectors method

Vector-lists method

Properties

Algorithms

Scalar multiplication

Hadamard product

Kronecker product

Common properties

Frobenius inner product

See also

References

External links

Ordinary matrix product