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MATLAB Basics With a brief review of linear algebra by Lanyi Xu modified by D.G.E. Robertson

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1. Introduction to vectors and matrices MATLAB= MATrix LABoratory What is a Vector? What is a Matrix? Vector and Matrix in Matlab

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What is a vector A vector is an array of elements, arranged in column, e.g., X is a n-dimensional column vector.

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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

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

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Vector addition Addition 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.

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Scalar multiplication A vector may be multiplied by a scalar, k, yielding

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Vector transpose The transpose of a vector is defined, such that, if x is the column vector its transpose is the row vector

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

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In addition, x T x, the squared length of the vector x, is The length or norm of vector x is denoted by

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Outer product of vectors The quantity of xy T is referred as the outer product and yields the matrix

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Similarly, we can form the matrix xx T as where xx T is called the scatter matrix of vector x.

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Matrix operations A 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

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Where a ij is the element in the i th row and j th column, for i=1,2, ,m and j=1,2,…,n. If m=2 and n=3, A is a 2 3 matrix

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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.

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Matrix addition Matrix 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.,

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For m=3 and n=n:

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

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For 3 2 matrix A,

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

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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 m p and B is p n, the product matrix [c ij ] has m rows and n columns, i.e.,

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For example, if A is a 2 3 matrix and B is a 3 2 matrix, then AB yields a 2 2 matrix, i.e., In general,

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For example, if and, then

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and Obviously,.

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

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

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Transpose of a matrix The transpose of a matrix is obtained by interchanging its rows and columns, e.g., if then Or, in general, A=[a ij ], A T =[a ji ].

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

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Inverse of a matrix In 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 that where I is an identity matrix.

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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.,

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It can be shown that

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MATLAB basic operations MATLAB is based on matrix/vector mathematics Entering matrices Enter an explicit list of elements Load matrices from external data files Generate matrices using built-in functions Create vectors with the colon (:) operator

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>> 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 >>

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

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>> I=eye(4)(i.e., identity matrix) I = 1 0 0 0 0 1 0 0 0 0 1 0 0 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 >>

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Generate Vectors with Colon (:) Operator The 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 > k j: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.

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

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Basic Permutation of Matrix in MATLAB sum, transpose, and diag Summation We can use sum() function. Examples, >> X=ones(1,5) X = 1 1 1 1 1 >> sum(X) ans = 5 >>

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>> 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 >>

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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 >>

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Expressions of MATLAB Operators Functions

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

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

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>> rho=(1+sqrt(5))/2 rho = 1.6180 >> a=abs(3+4i) a = 5 >>

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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.

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

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

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Distance between a Line and a Point given 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)

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