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4.1 BASICS

4.1 BASICS. Setting Up Matrices. Assignment a matrix A = [ {row 1}; {row 2}; ……; {row m} ] Ex: To set up a 4x4 matrix A ,like A = [1 1 1 1;1 2 3 4;1 3 6 10;1 4 10 20] It’s also legal to make input ‘look like’ output. A = [1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 ].

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4.1 BASICS

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  1. 4.1 BASICS

  2. Setting Up Matrices • Assignment a matrix A = [ {row 1}; {row 2}; ……; {row m} ] Ex: To set up a 4x4 matrix A ,like A = [1 1 1 1;1 2 3 4;1 3 6 10;1 4 10 20] • It’s also legal to make input ‘look like’ output. A = [1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 ]

  3. Simple Output • If you enter A = [1 2;3 4] without a semicolon ,then the system responds with • If you already assignment A wit a semicolon, but now you want to display it . You could only type A to display.

  4. Types and Dimension • Complex syntax c = complex(a,b), it means c = a + bi • Dimension is handled automatically in MATLAB. Suppose you set B = [1 2 3; 4 5 6 ], and then set B = [1 0; 0 1]. The system ‘knows enough’ to recognize that B has changed from 2-by-3 matrix to a 2-by-2 matrix.

  5. Subscripts • The subscripts in MATLAB must be positive. • If you type A(1.2 , 3.5) = 10, you will get a error message . ?? Subscript indices must either be real positive integers or logicals

  6. Vectors and Scalars • Vector and scalar are ‘special matrices’. The commands, x = [ 1 ; 2 ; 3 ] ; or x = [1 2 3]’ ; establish x as a column vector while • Assignment an 1-by-1 matrix with c = [3], and it equals to c = 3 .

  7. Size and Length If we declare A = [ 1 2 ; 3 4 ; 5 6 ], v = [7 8 9 10] • Size & Length syntax size(A) ans = [3 2] length(v) ans = 4

  8. Continuation • Sometimes it is not possible to put an entire MATLAB instruction on a single line. We can break the instruction in a ‘natural’ place. • There are some ways to express a instruction, and they are the same. A = [1 2 3; 4 5 6 ;....... 7 8 9] A = [1 2 3; 4 5 6 ; 7 8 9] A = [1 2 3; 4 5 6 ; 7 8 9]

  9. Variable Names • Variable names in MATLAB must consist of nineteen or fewer characters. • The name must begin with a letter.Any letter, number, underscore may follow, e.g., sym_part1. • Upper and lower cases are distinguished in MATLAB.

  10. Addition, Multiplication, Exponentiation • If A and B are matrices the C = A+B sets C to be the sum while C = A*B is the product. • To raise a square A to power r enter C = A^r • C = A^(-1) assigns the inverse of A to C. • f = c1Ab + c2A2b + c3A3b f = A*(A*(c(3)*A*b+c(2)*b)c(1)*b) f = ( c(3)*A^3+ c(2)*A^2+c(1)*A )*b

  11. Transpose • The conjugate transpose of a matrix can be obtained using a single quote: Set B = At B = A’

  12. The “Colon Method” for Setting Up Vectors • Colons can be used to prescribe vectors whose components differ by a fixed increment. • v = 1:3 v = [1 2 3] • v = 3:-1:0 v = [3 2 1 0] • v = 1:2:10 v = [1 3 5 7 9]

  13. Logarithmically Spaced Vectors • LOGSPACE logspace(a,b,n) generates n points between decades 10^a and 10^b. If w = logspace(d1,d2,n), then w(i) = 10 ^ [ d1 +(i-1)(d2-d1)/(n-1) ] • w = logspace(0,3,4) w = [1 10 100 1000]

  14. Generation of Special Matrices • MATLAB has a number of build-in functions that can be used to generate frequently occurring matrices: A = eye(m , n ) A m-by-n, ones on diagonal, zeros elsewhere. A = zeros(m , n ) A m-by-n, zeros everywhere. A = ones(m , n ) A m-by-n, ones everywhere.

  15. Random Matrices • It’s possible to generate matrices and vectors of random numbers: A = rand(m , n ) A m-by-n, random element between 0 and 1.

  16. Complex Matrices • If A and B are matrices with the same dimension and i2=-1, then the complex matrix C = A+iB can be generated as follows: C = A + sqrt(-1) * B ; or C = A + i * B ;

  17. Pointwise Operations • Element operations between two matrices of the same size are also possible: C = A.*B cij = aij * bij C = A.\B cij = bij / aij C = A./B cij = aij / bij C = A.^B cij = aijbij C = A.’ cij = aji • If we have a=[1 2], then type a.*[3 4] We get the answer = [1*3 2*4 ]= [3 8].

  18. More On Input/Output I • The format for printed output can be controlled using the following commands: format short 5-digit fixed point style format short e 5-digit floating point style format long 15-digit fixed point style format long e 15-digit floating point style format hex hexadecimal

  19. More On Input/Output II • format hex hexadecimal To represent a number into double precise floating point (with hexadecimal form). Ex. To show 1 in format ‘ hex ’ in matlab 1 = + 1.0 * 2^0 sign exponent+1023 mantissa 0011 1111 1111 0000 0000 0000 ..... 0000 3 f f 0 0 0 ..... 0 0+1023

  20. Machine Precision • At the start of MATLAB session, the variable eps contains the effective machine precision,i.e.,the smallest floating point number such that if x = 1 + eps then x>1 . • eps = 2.220446049250313e-016

  21. Flops • This is an obsolete function. • It can estimate the count of the program. • For complex operands, addition and subtraction count as two flops while multiplications and divisions each count as six flops.

  22. 4.2 LOOPS AND CONDITIONALS

  23. For-Loops • Syntax for {var} = {row vector of counter values} {statements} end • Ex: for i = 1:3 x(i) = i; end is equivalent to x = [1;2;3];

  24. Relations • Relation operators == equals ~= not equals < less than <= less than or equal to > greater than >= greater than or equal to • Ex: A=[1 2;3 4]; B=[1 2;2 4]; T = A==B Then, the output is T = 1 1 0 1

  25. ‘and’, ’or’, ’not’ • And(&), or(|), not(~) The operations are possible between 0-1 matrices of equal dimension. • If T1 = [1 1 ; 0 1] ,T2 = [1 0 ; 0 0] T=T1&T2, T= 1 0 0 0 T=T1|T2, T= 1 1 0 1 T=~T1, T= 0 0 1 0

  26. ‘if’ Constructions • Syntax if {relation} {statements} end • Ex:Set a upper triangular matrix that has 1’s on the diagonal and –1’s above the diagonal. for i = 1 : n for j = 1 : n if i < j A(i,j) = -1; elseif i > j A(i,j) = 0; else A(i,j) = 1; end end end

  27. The ‘any’ and ‘all’ Functions • Any If any component of the vector is nonzero, it returns a “1”. • All If all the entries are nonzero, it returns a “1”. • a = [ 0 1 1; 0 0 1; 0 0 1] any(a) ans =[0 1 1] all(a) ans =[0 0 1]

  28. While-Loops • Syntax while {relation} {statements} end • Print a sequence{Ai} of random 2-by-2 with the property that Ai+1>Ai: A = zeros(2) B = rand(2,2) while(B>A) A=B B=rand(2,2) end

  29. The ‘break’ Command • Break It can be used to terminate a loop. • Print all Fibonnaci number less than 1000: fib(1) = 1 fib(2) = 1 for j = 3:1000 z = fib(j-1) + fib(j-2) if z >= 1000 break end fib(j) = z end

  30. 4.3 WORKING WITH SUBMATRICES

  31. Setting Up Block Matrices • We can set up the block matrix as follows: A=[A11 A12 ; A21 A22 ; A31 A32 ] • If we want to make a 6-by-6 matrix with random 2-by-2 block diagonal, like A = rand(2,2) for i = 2:3 [m,m] = size(A); A = [ A zeros(m,2) ; zeros(2,m) rand(2,2) ] end

  32. The Empty Matrix • The empty matrix [] is often useful for initialization of certain matrix computations. B=[] B is empty • To set up a column-reversed version of above example: B = [] for j = 6:-1:1 B = [ B A(:,j) ]; end

  33. Designating Submatrices • If A is m-by-n matrix and i,j,p,q are integers. Then, B= A( i:j , p:q ) B= • B = A ( : , p:q ), the range ( : ) means from 1:n, if the dimension of matrix is n.

  34. Assignment to Submatrix If A = , we can assign the column 3 to [3 3 3]’ by using A(:,3)=3 or A ([7 8 9])=[3 3 3]. Then A =

  35. Kronecker Product • The Kronecker product of two matrices A and B can be calculated using the build-in function kron: C = kron(A,B) C = (aijB) • C = []; [m,n] = size(A); for j = 1:n ccol = []; for i = 1:m ccol = [ ccol ; A(i,j)*B ]; end C = [C ccol] end • If C = kron( [ 1 2 ] , [ 1 2 ; 3 4 ] ) then C = 1 2 2 4 3 4 6 8

  36. Turning Matrices into Vectors and Vice Versa • If A is a m-by-n matrix and we want to set v is an mn-by-1 vector. v = A(:) • Set a transpose of a 2-by-2 matrix A = [ 1 2 ; 3 4 ] x = A(:); x( [2 3] ) = x( [3 2] ); A(:) = x;

  37. 4.4 BUILD-IN FUNCTIONS

  38. ‘abs’, ‘real’, ‘imag’, ‘conj’ • B = abs(A) B = ( |aij| ) B = real(A) B = ( real(aij) ) B = imag(A) B = ( imag(aij) ) B = conj(A) B = real(A)-i*imag(A), i2=-1 • If A=[1-2i 2-i;3-4i 4-3i]; conj(A); Then, ans = 1+2i 2+i 3+4i 4+3i • MATLAB’s build-in functions are in lower case, unless user define. Otherwise, it’s will show ‘Capitalized internal function XXXX; Caps Lock may be on.’

  39. Norms • t = norm(A) t = || A ||2 t = norm(A,1) t = || A ||1 t = norm(A,’inf’) t = || A || t = norm(A,'fro') t = || A ||F • If A = [1 2 3 4] norm(A) ans = 5.47722557505166 sqrt(1+4+9+16) ans = 5.47722557505166

  40. Largest and Smallest Entries • If v = max(A) v = [ max(A(:,1)), … , max(A(:,n)) ] [v,i] = max(A) v = [ max(A(:,1)), … , max(A(:,n)) ] i = [the row of the max value in] • A =[ 1 2 8 6 9 4 7 5 3]; and [v,i]=max(A); v= 7 9 8 i = 3 2 1

  41. Sums and Products • t = sum(A) t = t = prod(A) t = • A =[ 1 2 9 6 9 4 7 5 3] sum(A) ans =14 16 15 prod(A) ans =42 90 96

  42. ‘sort’ and ’find’ I • Sorts each column of A in ascending order by using sort(A) • [v,i] = sort(A), returns in sorted v and an integer vector i which indicate the permutation of new from old matrix. • A =[ 3 5 9 2 6 8 1 4 7] [B,i]=sort(A); B =[1 4 7 2 5 8 3 6 9] i =[ 3 3 3 2 1 2 1 2 1]

  43. ‘sort’ and ’find’ II • The find function can be used to locate nonzero entries. • A =[1 0 2 3 0] find(A) ans =[1 3 4] find(A == 0) ans =[2 5]

  44. Rounding, Remainders, and Signs I • There are several conversion-to-integer routines: round(x) round x to nearest integer fix(x) round x to zero floor(x) round x to - ceil(x) round x to + • A = [ 1.1 -2.2 -0.9 3.8 ] round(A) ans=[1 - 2 -1 4] fix(A) ans=[1 -2 0 3] floor(A) ans=[1 -3 -1 3] ceil(A) ans=[2 -2 0 4]

  45. Rounding, Remainders, and Signs II • t = sign(x) t = • t = rem(x,y) t = x-y*fix(x/y) If y = 0, rem(x,0) is NaN. And x and y must be the same dimension. • rem equals mod(x,y) if x and y are the same sign. Otherwise , mod(x,y) = rem(x,y)+y a = mod(-7,4) a = 1 a = rem(-7,4) a = -3

  46. Extracting Triangular and Diagonal Parts I • B = tril (A) bij=aij if i j, zero otherwise. B = triu (A) bij=aij if i j, zero otherwise. B = tril (A,k) bij=aij if i+k j, zero otherwise. B = triu (A,k) bij=aij if i+k j, zero otherwise. • diag(v) puts v on the main diagonal. v=diag(A) return a vector v with A’s diagonal.

  47. Extracting Triangular and Diagonal Parts II • T = toeplitz(c,r) is equivalent to • T = zeros(n); for k = 1:n j = n-k; if k<n T = T + diag( r(k+1)*ones(j,1) , k); end T = T + diag( c(k)*ones(j+1,1) , -k+1); end

  48. Extracting Triangular and Diagonal Parts III • Take c=[6 7 8 9 10]; r=[1 2 3 4 5]; n=5; into above two function. We can show T =[ 6 2 3 4 5 7 6 2 3 4 8 7 6 2 3 9 8 7 6 2 10 9 8 7 6]

  49. ‘rank’ • If A is a m-by-n matrix, rank(A)= min{m,n}. • It Is also possible to base the rank decision on an arbitrary tolerance.If tol is a nonnegative scalar then r = rank(A , tol) • A = [ 1 2 ; 3 4 ; 5 6 ] rank(A)=min{3,2}=2 • A = [1 0.1 ; 3 0.2 ; 5 0.3 ] rank(A,1)=1

  50. Condition • cond(A) can be computed the 2-norm condition (the ratio of the largest singular value of X to the smallest) of a matrix A.

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