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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.

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VECTOR SYSTEM inequalities

A vector is a physical quantity characterizable by a module and a direction or orientation, which can be represented in polar coordinates by adding the orthogonal vector components parallel to the coordinate axes. Alternatively, a more formal and abstract, a vector is a physical quantity, which set a base, represented by a sequence of numbers or independent components such that their values \u200b\u200bare systematically relatable when measured by different observers.

Example

The distance between two cars

start from the same site can not be determined solely by its speed, this is, the modules of their speeds. If they are 30 and 40 km / h, to spend an hour, the distance between them may be, among other possibilities:

of 10 km, if the two cars are moving in the same direction.

From 70 km, if they move towards contraria.De 50 km, if they move in perpendicular directions.

Thus, the distance between the two cars depends not only on the speed of the cars (which marks the speedometer). It is necessary to define the rate vector character, ie the numerical data link address (or module).

scalars AND VECTORS

face those physical quantities such as mass, pressure, volume, energy

ed, temperature, etc., which are completely defined by a number and the units used in measurement, there are others, such as displacement, velocity, acceleration, force, the electric field, etc., which are not

c omplete
giving a numerical data set, but are associated with an address. The latter are called vector quantities as opposed to the first ones are called scalars.

The scalars are represented by the general m

simplest thematized, for a number. The vector quantities are represented by a mathematical entity called a vector. In a space Euclidean, not more than three dimensions, a vector is represented by a directed line segment. Thus, a vector is characterized by the following elements: its length, or module, always positive by definition, and its direction determined by the angle between the vector with the coordinate axes. Thus we can state:

A vector is a physical quantity that has magnitude and direction.

is represented as a directed line segment, with an address, drawn in a similar way "arrow." Its length is the mod

Ulo

vector and "Arrowhead" indicates its direction, which is measured in polar coordinates.

NOTATION

The vector quantities are represented in the printed texts of letters in bold to distinguish them from scalar quantities that are represented in italics. In the manuscripts, the vector magnitudes are represented by placing an arrow on the letter that identifies your module (Which is a scalar). Examples:
... represent, respectively, the vector magnitudes of modules A, a, ω, ... The module is also a vector quantity is enclosed in slashes for the vector notation: ... In the manuscripts
write: ... for vectors and ... or ... for modules.

Where appropriate, we represent the vector magnitude reference to the origin and the end of the line segment that represents geometrically so, denote the vectors represented in Figure 2 on the way ... This notation is very useful for vectors displacement.

addition to these conventions or vestors unit vectors, whose magnitude is unity, are represented with a circumflex over frequently.

TYPES OF VECTORS

According to the criteria used to determine equality or two equipotent vectors, we can distinguish different types of them:

Vectors free: they are not applied at any point in particular.Vectores sliding: its point of application can slide along the line of action. Vector
fixed or linked, are applied in a particular point.

We also refer to:

unit vectors: unit vectors module. Concurrent Vector
: its lines of action converge at a point proper or improper (parallel). Vector
opposite: vectors of equal magnitude but opposite direction. Collinear vectors
: sees

ctor that company

rten same line of action.
coplanar vectors: the vectors which are coplanar lines of action (located in the same plane)

COMPONENTS OF A VECTOR

A vector space can be expressed as a linear combination of three unit vectors perpendicular or vestors to provide a basis vector.
In Cartesian coordinates, the are represented by unit vectors i, j, k, parallel to the axes of coordinates x, y, z positive. The components of the vector into a default base vector can be written in parentheses and separated by commas:
or expressed as a combination of unit vectors defined in the base vector. Thus, in a Cartesian coordinate system will be otherwise.

These representations are equivalent to each other, and the values \u200b\u200bx, y, z are the vector components, unless otherwise indicated, are real numbers.

OPERATIONS WITH VECTORS

Vector Addition:

To add two vectors and vector free vector are chosen as representatives of two vectors such that the tail end of a match with the source end of another vector.


parallelogram method :

is to have two vectors graphically so that the origins of both agree on one point, completing a parallelogram by drawing lines parallel to each of the vectors in the extr

emo the other (see graphic at right). The result of the sum is the diagonal of the parallelogram of the common origin of both vectors.

triangle method:

is to have a vector graphic after another, ie the origin of a vector is carried over the end of another. Here are links the origin of the first vector with the end of the second.

analytical method. Sum and difference vectors:

Given two free vectors,

The result of the sum or the difference is expressed how

and ordering the components,

Known
modules given two vectors a and b and the angle θ formed between itself, the module is:

The derivation of this expression can be found on deduction of the sum module.
product of a vector by a scalar

PRODUCT OF A VECTOR BY A SCALAR

The product of a vector by a scalar is another vector whose magnitude is the scalar product of the magnitude of the vector, whose direction is equal to the vector, or against it if the scale is negative.

Based on the graphical representation of the vector, on the same line so many times we address the vector form as check scaling.

Based on a scale, "N" and a vector "A", the product of "N" to "A" is represented and is done by multiplying each of the components of the vector by the scalar that is, given the vector

your product is the scalar

that is, multiplied by "N" each of the components vector.

DERIVATIVE OF A VECTOR

Given an array is a function of independent variable

estimate the derivative of the vector with respect to variable t, by calculating the derivative of each of its scalar components as if it were:

taking note that the unit vectors are constant in magnitude and direction.

An example of derivation of a vector, based on a vector function:

This function represents a helical curve around the z axis, of unit radius, as illustrated in Fig. We can imagine that this curve is the trajectory of a particle and the function represents the position vector function of time t. Differentiating we have:

Performing the derivative:

The derivative of the position vector with respect to time is speed, so this second function determines the particle velocity vector versus time, we can write:

This velocity vector is a vector tangent to the trajectory at the point occupied by the particle at every moment. If derivásemos again would get the acceleration vector.

angle between two vectors

The angle determined by the directions of two vectors A and B is given by: