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MATRICES AND DETERMINANTS

In mathematics, a matrix is \u200b\u200ba table of numbers consisting of abstract quantities that can be added and multiplied. The matrices are used to

describe systems of linear equations, keep track of the coefficients of a linear and record data that depend on various parameters. Arrays are described in the field of matrix theory. Can add, multiply and decompose in various ways, which also makes a key concept in the field linear algebra.

Definitions and notations

A matrix is \u200b\u200ba square or rectangular table of data (called elements or entries in the matrix) arranged in rows and columns, where each row is the horizontal lines of the matrix and a column is each of the vertical lines. A matrix with m rows and n columns is called matrix m-by-n (written m × n)-dimensional matrix yamyn. The dimensions of a matrix are always given with the number of rows first and the number of columns later. It is commonly said that a matrix m-by-n has an order of m × n ("order" has the meaning of size.) Two matrices are said to be equal if they are of the same order and have the same elements. When an array element is in the ith row and jth column is called element i, jo (i, j)-ith matrix. Put back rows first and then columns.

Almost always, the matrices are denoted with capital letters while the corresponding letters are used lower case to denote the same elements. For example, a matrix element found in the ith row and jth column is denoted as ai, joa [i, j]. Alternative notations are A [i, j] or Ai, j. In addition to using capital letters to represent matrices, many authors represent matrices with bold font to distinguish them from other types of variables. Thus A is a matrix, while A is a scalar.

normally written to define an m × n matrix with each entry in the matrix 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) is called a row vector and a matrix m × 1 (column m rows) is called vector column.

Example:

matrix,

is
4x3 matrix. The element A [2.3] or a2, 3 is 7.

matrix,

is a matrix 1 × 9, or a row vector with 9 elements.

BASIC OPERATIONS

Sum or addition: 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 homologous elements of the matrices to add. For example:

PROPERTIES

Associations:

Given the m × n matrices A, B and C
A + (B + C) = (A + B) + C

Commutative:

Given m × n matrices A and B
A + B = B + A

Existence of zero matrix or zero matrix:

A + 0 = 0 + A = A matrix

Existence opposite:
with gr-A = [-aij]
A + (-A) = 0

SCALAR PRODUCT BY

Given a matrix A and a scalar c, the product cA is calculated by multiplying the scalar by each element of A (ie (cA) [i, j] = cA [i, j]).

Example:

PROPERTIES

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

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 an m × n matrix B is a matrix n × p, then their matrix product AB is m × p matrix (m rows, p columns) given by:

for each pair i and j .

For example:

SQUARE MATRICES AND RELATED DEFINITIONS

A square matrix is \u200b\u200ba matrix that has the same number of rows and columns. The set of all n-square matrices by-n with addition and multiplication of matrices is a ring that is not generally commutative.
M (n, R), the ring of real square matrices is an algebra real associative unit. M (n, C), the ring of complex square matrices, is a complex associative algebra.
In
The identity matrix of order n is n by n matrix in which all principal diagonal elements are equal to 1 and all other elements equal 0. The identity matrix is \u200b\u200bso named because it satisfies the equations and InN MIn = M = N for any matrix M m N n by n and k. For example, if n = 3:

The identity matrix is \u200b\u200bthe unit element in the ring of square matrices.
invertible elements of this ring are called invertible matrices or nonsingular matrices. An n by n matrix A is invertible if and only if there exists a matrix B such that AB = In =
BA.

In this case, B is the inverse matrix of A, identified by A-1. The set of all invertible n by n matrices forms a group (specifically a Lie group) under matrix multiplication, the general linear group.

If λ is a number v is a null vector such that Av = λv, then we say that v is an eigenvector of A and λ is its associated eigenvalue. The number λ is an eigenvalue of A if and only if A-λIn is not invertible, what happens if and only if pA (λ) = 0, where pA (x) is the characteristic polynomial of A. pA (x) is a polynomial of degree n and therefore has no complex roots if multiple roots are counted according to multiplicity. Every square matrix has at most n complex eigenvalues.

The determinant of a square matrix is \u200b\u200bthe product of its n eigenvalues, but can also be defined by the Leibniz formula. Invertible matrices are precisely those matrices whose determinant is nonzero.

Gaussian elimination algorithm can be used to calculate the determinant, rank and the inverse of a matrix and for solving systems of linear equations.
The trace of a square matrix is \u200b\u200bthe sum of the diagonal elements, which is the sum of its n eigenvalues.