What Times What Equals 51 – “Matrix Theory” redirects here. For the physics topic, see Matrix theory (physics). For other uses of “matrix”, see Matrix (disambiguation).
M × n matrix: m rows are horizontal and n columns are vertical. Each matrix element is usually separated by two subscript variables. For example, a2 rearranges the 1 element to the second row and first column of the matrix.
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A matrix (plural matrix) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, used to represent mathematical objects or properties of such objects.
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Is a matrix with two rows and three columns. It is often called a “two-to-three matrix”, “2 × 3 matrix”, or 2 × 3 reduced matrix.
Without further definition, matrices allow us to rework linear maps and perform some computations on linear algebra. Therefore, the study of matrices is a major part of linear algebra, and most properties and operations of abstract linear algebra can be represented by matrices. For example, matrix multiplication reproduces the linear map structure.
Not all matrices are relevant to linear algebra. This is especially the case for graph theory, incidence matrices and adjacency matrices.
This article will focus on matrices related to linear algebra, and unless otherwise noted, all matrices reproduce linear maps or can be viewed as such.
Square matrices and matrices with the same number of rows and columns play a central role in matrix theory. The squares of the given dimensions form an invariant circle, which is one of the most common examples of invariant circles. The determinant of a square matrix is a number associated with the matrix, which plays an important role in the study of square matrices. For example, a square matrix is invertible only if it has a nonzero determinant, and the eigenvalue of the square matrix is the root of the determinant of the polygon.
In geometry, matrices are widely used to describe and retrieve geometric transformations (eg, rotations) and coordinate transformations. In numerical analysis, many computational problems are solved by reducing them to matrix computations, which often require the computation of large fuzzy matrices. Matrices are used in most branches of mathematics and most branches of science, either directly or by use in geometry and numerical analysis.
Matrix theory is a branch of mathematics that focuses on the study of matrices. It was initially a subfield of linear algebra, but soon included topics related to graph theory, algebra, combinatorics, and statistics.
A matrix is a rectangular array of numbers (or other mathematical objects) called a matrix trie. Matrices are subject to standard operations such as addition and multiplication.
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Real matrices and complex matrices are matrices that contain real numbers or complex numbers. More test types are discussed below. For example, this is a real matrix:
The numbers, letters, or expressions in a matrix are called its elements or elements. The horizontal and vertical lines of test in a matrix are called rows and columns.
The size of a matrix is determined by the number of rows and columns it contains. There is no limit to the number of rows and columns when the matrix (in general sse) is a positive integer. A matrix with m rows and n columns is called an m × n matrix or m-matrix, while m and n are called its dimensions. For example, the matrix A above is a 3 × 2 matrix.
A matrix with one row is called a row vector, and a matrix with one column is called a column vector. A matrix with the same number of rows and columns is called a square matrix.
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A matrix with an infinite number of rows and columns (or both) is called an infinite matrix. In some situations, such as computer algebra programs, it is useful to consider a matrix that has no rows or columns, called an empty matrix.
A matrix, with the same number of rows and columns, is sometimes used to reproduce linear transformations from vector space to itself, such as reflections, rotations, and shears.
The marked features of the symbol matrix are very diverse, some of which are dominant trds. Matrices are often written in square brackets or fractional form, so an m × n mtimes n matrix is rewritten as A mathbf.
Or A = ( a i , j ) 1 ≤ i , j ≤ n =(a_)_} in case n = m n = m.
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A matrix is usually denoted by an uppercase letter (like A in the example above) and the corresponding lowercase letter with two subscripts (like:
), Try again. In addition to using capital letters to denote matrices, many authors use a special typographical style to distinguish matrices from other mathematical objects, often with vertical (rather than italic) bold letters. An alternative notation is to use double underscores with variable names in bold or without bold, as in A _ _ }}.
A try in row i and column j of a matrix A is sometimes called a matrix i, j or (i, j) try, and is usually separated by a.
A = [ 4 − 7 5 0 − 2 0 11 8 19 1 − 3 12 ] =4&-7&color &0\-2&0&11&8\19&1&-3&12d}}
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= f(i, j). For example, each try of the following matrix A is defined by the formula aij = i − j.
A = [ 0 − 1 − 2 − 3 1 0 − 1 − 2 2 1 0 − 1 ] =0&-1&-2&-3\1&0&-1&-2\2&1&0&-1d}}
In this case, the matrix is sometimes defined by a formula inside square brackets or doubles. For example, the above matrix is defined as A = [i−j], or A = ((i−j)). If the size of the matrix is m × n, then the above formula f(i, j) is true for any i = 1, …, m and any j = 1, …, n. It can be specified separately, or m × n can be specified as a subscript. For example, the matrix A above is 3 × 4, defined as A = [i − j] (i = 1, 2, 3; j = 1, …, 4) or A = [i − ” can be done as . J]
Some programming languages use duplicate arrays (or arrays of arrays) to reconstruct m-by-n matrices. Some programming languages initialize array indices to zero, in which case m arrays are indexed by 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1.
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Asterisks are sometimes used to denote entire rows or columns of a matrix. For example, A
The set of all m-by-n real matrices is usually the point M ( m , n ) , }(m, n) } or M m × n ( R ). __(mathbb ).} The sum of all m by n matrices over a distinct field or ring R is M ( m , n , R ), } (m, n, R), } or dot. M m × n ( R ) . __(ر) }_(R).}  M is often used instead of M. }.
There are several basic operations that can be used to transform matrices, and they are called matrix addition, scalar multiplication, transposition, matrix multiplication, string operations, and submatrix operations.
[ 1 3 1 1 0 0 ] + [ 0 0 5 7 5 0 ] = [ 1 + 0 3 + 0 1 + 5 1 + 7 0 + 5 0 + 0 ] = [ 1 3 6 8 5 0 ] 1&3&1\ 1&0&0d}+0&0&5\7&5&0d}=1+0&3+0&1+5\1+7&0+5&0+0d}=1&3&6\8&5&0d}}
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The product of a number c (called a scalar in abstract algebra terms) and a matrix cA is computed by multiplying each attempt of A by c.
This operation is called scalar multiplication, but “scalar product” is sometimes synonymous with “inner product”, so the result is not called “scalar product” to avoid confusion.
2 ⋅ [ 1 8 − 3 4 − 2 5 ] = [ 2 ⋅ 1 2 ⋅ 8 2 ⋅ − 3 2 ⋅ 4 2 ⋅ − 2 2 ⋅ 5 ] = [ 2 16 − 6 1 ] = [ 2 16 − 6 1 ] & 8 4&-2&5d}=2cdot 1&2cdot 8&2cdot -3\2cdot 4&2cdot -2&2cdot 5d}=2&16&-6\8&-4&10d} }
Familiar properties of numbers that apply to these matrix operations: For example, addition is invariant, meaning that the sum of matrices does not depend on the order of the sums: A + B = B + A.
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The product of two matrices is defined only if the number of columns of the left matrix is equal to the number of rows of the right matrix. If A is an m-matrix and B is an n-matrix, then their matrix product AB is the m-matrix given by the dot product of A and the corresponding row.
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