MATLAB BasicsWith a brief review of linear algebraby Lanyi Xumodified by D.G.E. Robertson

1. Introduction to vectors and matricesMATLAB= MATrix LABoratoryWhat is a Vector?What is a Matrix?Vector and Matrix in Matlab

What is a vectorA vector is an array of elements, arranged in column, e.g.,

X is a n-dimensional column vector.

In physical world, a vector is normally 3-dimensional in 3-D space or 2-dimensional in a plane (2-D space), e.g.,

, or

If a vector has only one dimension, it becomes a scalar, e.g.,

Vector additionAddition of two vectors is defined by

Vector subtraction is defined in a similar manner. In both vector addition and subtraction, x and y must have the same dimensions.

Scalar multiplicationA vector may be multiplied by a scalar, k, yielding

Vector transposeThe transpose of a vector is defined, such that, if x is the column vector

its transpose is the row vector

Inner product of vectorsThe quantity xTy is referred as the inner product or dot product of x and y and yields a scalar value (or x y).If xTy = 0x and y are said to be orthogonal.

In addition, xTx , the squared length of the vector x , is

The length or norm of vector x is denoted by

Outer product of vectorsThe quantity of xyT is referred as the outer product and yields the matrix

Similarly, we can form the matrix xxT as

where xxT is called the scatter matrix of vector x.

Matrix operationsA matrix is an m by n rectangular array of elements in m rows and n columns, and normally designated by a capital letter. The matrix A, consisting of m rows and n columns, is denoted as

Where aij is the element in the ith row and jth column, for i=1,2,,m and j=1,2,,n. If m=2 and n=3, A is a 23 matrix

Note that vector may be thought of as a special case of matrix: a column vector may be thought of as a matrix of m rows and 1 column; a rows vector may be thought of as a matrix of 1 row and n columns;A scalar may be thought of as a matrix of 1 row and 1 column.

Matrix additionMatrix addition is defined only when the two matrices to be added are of identical dimensions, i.e., that have the same number of rows and columns.

e.g.,

For m=3 and n=n:

Scalar multiplicationThe matrix A may be multiplied by a scalar k. Such multiplication is denoted by kA wherei.e., when a scalar multiplies a matrix, it multiplies each of the elements of the matrix, e.g.,

For 32 matrix A,

Matrix multiplicationThe product of two matrices, AB, read A times B, in that order, is defined by the matrix

The product AB is defined only when A and B are comfortable, that is, the number of columns is equal to the number of rows in B. Where A is mp and B is pn, the product matrix [cij] has m rows and n columns, i.e.,

For example, if A is a 23 matrix and B is a 32 matrix, then AB yields a 22 matrix, i.e.,In general,

For example, ifand, then

andObviously,.

Vector-matrix ProductIf a vector x and a matrix A are conformable, the product y=Ax is defined such that

For example, if A is as before and x is as follow,, then

Transpose of a matrixThe transpose of a matrix is obtained by interchanging its rows and columns, e.g., if thenOr, in general,A=[aij], AT=[aji].

Thus, an mn matrix has an nm transpose.For matrices A and B, of appropriate dimension, it can be shown that

Inverse of a matrixIn considering the inverse of a matrix, we must restrict our discussion to square matrices. If A is a square matrix, its inverse is denoted by A-1 such thatwhere I is an identity matrix.

An identity matrix is a square matrix with 1 located in each position of the main diagonal of the matrix and 0s elsewhere, i.e.,

It can be shown that

MATLAB basic operationsMATLAB is based on matrix/vector mathematicsEntering matricesEnter an explicit list of elements Load matrices from external data files Generate matrices using built-in functionsCreate vectors with the colon (:) operator

>> x=[1 2 3 4 5];>> A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1]

A = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1>>

Generate matrices using built-in functionsFunctions such as zeros(), ones(), eye(), magic(), etc. >> A=zeros(3)A = 0 0 0 0 0 0 0 0 0>> B=ones(3,2)B =1 11 11 1

>> I=eye(4)(i.e., identity matrix)I =1 0 0 00 1 0 00 0 1 00 0 0 1>> A=magic(4) (i.e., magic square)A =16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1>>

Generate Vectors with Colon (:) OperatorThe colon operator uses the following rules to create regularly spaced vectors:

j:k is the same as [j,j+1,...,k]j:k is empty if j > kj:i:k is the same as [j,j+i,j+2i, ...,k]j:i:k is empty if i > 0 and j > k or if i < 0 and j < k

where i, j, and k are all scalars.

Examples>> c=0:5c = 0 1 2 3 4 5>> b=0:0.2:1b = 0 0.2000 0.4000 0.6000 0.8000 1.0000>> d=8:-1:3d =8 7 6 5 4 3>> e=8:2e = Empty matrix: 1-by-0

Basic Permutation of Matrix in MATLABsum, transpose, and diagSummationWe can use sum() function.Examples,>> X=ones(1,5)X = 1 1 1 1 1>> sum(X)ans =5>>

>> A=magic(4)A = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

>> sum(A)ans = 34 34 34 34

>>

Transpose>> A=magic(4)A = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

>> A'ans = 16 5 9 4 2 11 7 14 3 10 6 15 13 8 12 1

>>

Expressions of MATLABOperatorsFunctions

Operators+Addition-Subtraction*Multiplication/Division\Left division^Power' Complex conjugate transpose( )Specify evaluation order

FunctionsMATLAB provides a large number of standard elementary mathematical functions, including abs, sqrt, exp, and sin. pi3.14159265...iImaginary unit ( )jSame as iUseful constants:

>> rho=(1+sqrt(5))/2rho = 1.6180

>> a=abs(3+4i)a = 5

>>

Basic Plotting Functions plot( )The plot function has different forms, depending on the input arguments.

If y is a vector, plot(y) produces a piecewise linear graph of the elements of y versus the index of the elements of y.

If you specify two vectors as arguments, plot(x,y) produces a graph of y versus x.

Example,

x = 0:pi/100:2*pi;y = sin(x);plot(x,y)

Multiple Data Sets in One Graphx = 0:pi/100:2*pi;y = sin(x);y2 = sin(x-.25);y3 = sin(x-.5);plot(x,y,x,y2,x,y3)

Distance between a Line and a Pointgiven line defined by points a and b find the perpendicular distance (d) to point c

d =

norm(cross((b-a),(c-a)))/norm(b-a)