Matrix (mathematics)
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In mathematics , an array is a rectangular management numbers or, more generally, a table consisting of abstract quantities can add and multiply .
Table of Contents [hide ]
1 Definitions and notations
2 Example 3 Sum of matrices
3.1 Properties of matrix addition
4 Product of a matrix by a scalar
4.1 Properties of the Scalar Product 5 Product
matrix array 6 Division 7
inverse matrix
8 Classes of matrices
9 Arrays in Computer History
10 11 Notes
12 See also
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Definitions and notations [edit ]
A table or matrix is \u200b\u200ba rectangular array of numbers. The numbers in the array are called array elements.
The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called matrix m-by-n (written m × n), m and n are its dimensions. The dimensions of a matrix are always given the number of rows first and the number of columns later.
input matrix A found in the ith row and j-th column is called input i, jo entry (i, j)-ith of A. This is written as Ai, jo A [i, j].
normally written to define an m × n matrix with each entry in the array A [i, j] called aij for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. However, the convention the start of the indices i and j in 1 is not universal: some programming languages \u200b\u200bstart at zero, in which case one has 0 ≤ i ≤ m - 1 and 0 ≤ j ≤ n - 1.
A matrix with one column or one row is often called a vector, and is interpreted as an element of Euclidean space . 1 × n matrix (one row and n columns) called row vector and a matrix m × 1 (one column and m rows) is called a column vector . Example
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matrix is \u200b\u200ba 4 × 3 matrix. The element A [2.3] or a2, 3 is 7.
matrix is \u200b\u200ba matrix 1 × 9, or a row vector with 9 elements.
Sum of matrices [edit ]
Given the matrices m-by-n A and B, their sum A + B is the matrix m-by-n calculated by adding the corresponding elements (ie (A + B) [i, j] = A [i, j] + B [i, j]). Ie adding each of the counterparts of the matrices elemetos add. For example:
Properties of the sum of matrices [edit ]
Associative arrays
Given m-by-n A, B and C
A + (B + C) = (A + B) + C
Commutative
Given the matrices m-by-n A and B
A + B = B + A
Existence of zero matrix or zero matrix
A + 0 = 0 + A = A matrix Existence opposite
with-A = [-aij ]
A + (-A) = 0
product of a matrix by a scalar [edit ]
Given a matrix A and a number c, cA scalar product is calculated by multiplying the scale c for each element of A (ie (cA) [i, j] = cA [i, j]). For example:
Scalar Product Properties [edit ]
Let A and B matrices and c and d scalars.
Closure: If A is matrix and c is scalar, then cA is the parent.
Associativity: (cd) A = c (dA)
Neutral Element: 1 • A = A
Distributivity:
of scale: c (A + B) = cA + cB
matrix: (c + d) A = cA + dA
product matrices [edit ]
Main article: Matrix Product
The product of two matrices can be defined only if the number of columns in the left matrix is \u200b\u200bthe same as the number of rows right matrix. If A is a matrix m-by-n matrix and B is an n-by-p, then their matrix product AB is the parent m-by-p (m rows, p columns) given by:
for each pair i and j.
For example:
The product of two matrices is not commutative, ie AB ≠ BA. The division between matrices, ie the operation that could produce the ratio A / B, is not defined. However, there is the concept of inverse matrix , applicable only to square matrices
