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Getting started with MATLAB
Written by Burak Ozturk
Last revision: January 18, 2001
Some introductory example problems with solutions:
To start, let’s see how some simple matrix calculations are performed using MATLAB.
Verify the following matrix properties:
1. If [A] is symmetric, the matrix [D] = [A] [B] is, in general, not symmetric, even
if [B] is also symmetric.
2. If [A] is symmetric, the matrix [D] = [B] T^ [A] [B] is always symmetric.
3. If [A] [B] = [C] [B], it does not necessarily follow that [A] = [C].
4. If [D] = [A] [B], then [D] T^ = [B] T^ [A] T
5. det [A] = Π λj where λj is the j-th eigenvalue of [A] (Use help in MATLAB to
see the commands det, prod, eig)
Answers: Prob 1:
A=[2 4 1;4 8 2;1 2 3] A = 2 4 1 (^41 82 )
B=[4 6 3;6 1 2;3 2 8] B = (^46 61 ) 3 2 8 D=A*B D = (^3570 1836 ) 25 14 31 % A and B are both symmetric matrices, where their product, which is D,% is not a symmetric matrix.
Prob 2:
% Using the same matrices A and B from the problem 1.1;
D=B'AB D = (^635330 330172 ) 445 238 402 % The matrix D=B'AB is symmetric.
Prob 3:
B=[0 0;0 0];>> A=[1 2;3 2]; C=[4 4;5 2]; AB ans = 0 0 0 0 CB ans = (^00 )
% Here, AB=CB, although A and C are different matrices.
Prob 4:
A=[5 7;3 2];>> B=[1 3;4 4]; D=A*B; A = (^53 )
B = (^14 )
D = 33 11 4317
D' ans =
% B= any matrix of size m x n % Output variable: % A= a matrix of size n x m [m n]=size(B); for i=1:mfor j=1:n
end A(j,i)=B(i,j); end
% Now we are going to check the function where we let it operate on a % random matrix B of size 4x6. % >> B=rand(4,6) % B = %% 0.95010.2311 0.89130.7621 0.82140.4447 0.92180.7382 0.93550.9169 0.05790. % 0.6068 0.4565 0.6154 0.1763 0.4103 0. % 0.4860 0.0185 0.7919 0.4057 0.8936 0. % >> [A]=newtransp(B) % A = %% 0.95010.8913 0.23110.7621 0.60680.4565 0.48600. % 0.8214 0.4447 0.6154 0. %% 0.92180.9355 0.73820.9169 0.17630.4103 0.40570. % 0.0579 0.3529 0.8132 0.
% You can see that the output matrix A is the transpose of the % input matrix B. The function operates correctly.
2. Compute the cross product: Write a function called “crossprod” to compute the
cross product of two vectors u, v , given u = (u 1 , u 2 , u 3 ), v=(v 1 , v 2 , v 3 ). Check your
function by taking cross products of pairs of unit vectors: (i,j), (j, k), etc.
i={1, 0, 0}T, j={0, 1, 0}T, k={0, 0, 1}T.
You can also test your function using the MATLAB built-in function “cross”
here.
Answer:
function [w]=crossprod(u,v) % Input variables % Two vectors u=(u1,u2,u3) and v=(v1,v2,v3) % Output variable is the vector w=(w1,w2,w3), which is the cross % product of the vectors u,v. w(1,1)=u(2,1)v(3,1)-u(3,1)v(2,1); w(2,1)=u(3,1)v(1,1)-u(1,1)v(3,1);w(3,1)=u(1,1)v(2,1)-u(2,1)v(1,1);
% Now we want to check the function by taking cross products of pairs% of unit vectors.
%% >> i=[1;0;0];>> j=[0;1;0]; % >> k=[0;0;1];
% >> [w]=crossprod(i,j) % w = %% (^00) % 1 % The output vector w is equal to the unit vector k=[0;0;1], which % shows that the function works correctly.
% Similarly; % >> [w]=crossprod(j,k) % w = % 1 %% (^00)
% >> [w]=crossprod(i,k) % w = % 0 %% -1 0
% Now we call the function;
[P]=matmul(K,d) P = 148 129 142 56 flops ans = 24
% For checking purposes, we simply calculate the product of our two % matrices K and d.
K*d ans = (^148142 )
flops ans = 24 % Multiplying a 2x3 matrix K with a 3x2 matrix d resulted in a 2x2% matrix P, as expected.
% The result we get using the program we wrote and the result we get by% simply multiplying the two matrices K and d are the same. So the % program works correctly and also in a most efficient way by % performing 24 flops, which is exactly the same number of flops the% MATLAB built-in arithmetic operation uses to perform the same % calculation.
