Question

In the next exercises, we consider square n X n matrices; I is the identity matrix (In MATLAB eye (n) gives a square n by n matrix). If ex is the column unit vector which components are all O's except the kth component which is equal to 1, i.e., 이".e1 = 101 0 then the identity matrix I is such that: 10 000 ei = [100 01T, , el 1000 11T. 0 0 T' 0 0 1 To generate in MATLAB the unit (colunn) vector er e R", one may uses the sequence of commands: I-eye (n);e(1)-I:,i) Exercise 1.5 Let A Rnxn, r E Rk. Find the first column of using a sequence of GAXPY's operations

Answer 1

The column version of GAXPY must be applied here.(Since x is a column vector according to the definition). The following code does this.

clc
close all
clear all

%Input matrix of size of n x n
% A = [1 2 3;4 5 6;7 8 9];
A = ones(5,5)
% A=round(20*rand(5,5))   % RAndom A generation
n=size(A,2);
%generation of x; x is a column vector
%x1 = I(:,1)===> x1 = [1 0 0]'
%Each column of I represents x1,x2....xn

I=eye(n)

%computing (A-xkI) terms using GAXPY column version
for i=1:n
    for j=1:n
        x=I(:,i); %each time column of I is stored in x as in your question
        I_up{i}=A-x(j)*I; %single term computation
    end
end

%multiplication of obtained matrices for getting M
M=1;
for i=1:length(I_up)
    M=M*I_up{i};
end

%first column of M
disp('First Column of M')
disp(M(:,1))

Result of the code:

$MATLAB 7.8.0 File Edit Debug Parallel Desktop Window Help 。e3 |美9a喒つ 都ゴヨ1 @ Current Directory:CAUsersUSERDocuments\MATLAB Shortcuts a How to AddWhats New A - Square Matrix 0 100Each column of I is:x ist Column o S00 500 S00 500 S00 result 兵>>$