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The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]
Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).
the algebra of all n-by-n matrices over a field (or commutative ring) K. Here the multiplication is ordinary matrix multiplication. group algebras, where a group serves as a basis of the vector space and algebra multiplication extends group multiplication. the commutative algebra K[x] of all polynomials over K (see polynomial ring).
In other words, the scalar multiplication of ¯ satisfies = ¯ where is the scalar multiplication of ¯ and is the scalar multiplication of . The letter v {\displaystyle v} stands for a vector in V , {\displaystyle V,} α {\displaystyle \alpha } is a complex number, and α ¯ {\displaystyle {\overline {\alpha }}} denotes the complex conjugate ...
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. [1] 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 ...
Vector addition is just matrix addition and scalar multiplication is defined in the obvious way (by multiplying each entry by the same scalar). The zero vector is just the zero matrix. The dimension of F m×n is mn. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries 0.
An identity matrix of any size, or any multiple of it is a diagonal matrix called a scalar matrix, for example, []. In geometry , a diagonal matrix may be used as a scaling matrix , since matrix multiplication with it results in changing scale (size) and possibly also shape ; only a scalar matrix results in uniform change in scale.
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