Linear Algebra/Matlab Assignment Help Needed (*Requires Matlab software) Let A =

Question

Linear Algebra/Matlab Assignment Help Needed (*Requires Matlab software)

Let A = [14 24 14 -31 45 31 12 26 14 -36 40 26 38 76 48 -104 140 94 12 24 14 -34 40 26 -19 -38 -23 52 -68 -47 7 14 9 -21 25 18] (a) Using the eig command, give the distinct eigenvalues of A and their multiplicities. (b) For each distinct eigenvalue lambda_j you found in part (a), use rref(lambda_j I - A) to find a basis for the eigenspace associated with lambda_j. Do not use the Matlab command [X.D] = eig (A). (c) Using the results of (b), give the dimension of each eigenspace. (d) Using the results of (a) and (c), determine whether A is diagonalizable. (e) If A is diagonalizable, use the results of (b) to find a nonsingular matrix P such that P^-1 AP is a diagonal matrix, and then check your work by using MATLAB commands to compute the matrix B = P^-1 AP.

Explanation / Answer

MATLAB code:

clear
clc

A = [14,12,38,12,-19,7;
     24,26,76,27,-38,14;
     14,14,48,14,-23,9;
     -31,-36,-104,-34,52,-21;
     45,40,140,40,-68,25;
     31,26,94,26,-47,18];
E = eig(A);
eigen_spaces = [];
for i=1:length(E)
    lambda = E(i);
    eigen_spaces = [eigen_spaces;rref(lambda*eye(size(A,1))-A)];
end
[~,P] = eig(A);
D = inv(P)*A*P;

Output:

>> A
A =
    14    12    38    12   -19     7
    24    26    76    27   -38    14
    14    14    48    14   -23     9
   -31   -36 -104   -34    52   -21
    45    40   140    40   -68    25
    31    26    94    26   -47    18
>> E
E =
   0.0564 + 9.8829i
   0.0564 - 9.8829i
-3.0000 + 0.0000i
   2.8871 + 0.0000i
   2.0000 + 0.0000i
   2.0000 - 0.0000i
>> P
P =
Columns 1 through 3
   0.0564 + 9.8829i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0564 - 9.8829i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i -3.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
Columns 4 through 6
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   2.8871 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   2.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   2.0000 - 0.0000i
>> D
D =
   1.0e+02 *
Columns 1 through 3
   0.1400 + 0.0000i -0.1200 - 0.0014i -0.0007 + 0.1153i
-0.2400 + 0.0027i   0.2600 - 0.0000i -0.0013 - 0.2307i
-0.0026 - 0.4612i -0.0026 + 0.4612i   0.4800 + 0.0000i
-0.0061 - 1.0611i -0.0070 + 1.2323i   1.0807 + 0.0000i
   0.0127 + 2.2236i   0.0113 - 1.9766i -2.1000 + 0.0000i
   0.0087 + 1.5318i   0.0073 - 1.2848i -1.4100 - 0.0000i
Columns 4 through 6
   0.0002 - 0.0351i -0.0002 + 0.0384i   0.0001 - 0.0142i
   0.0005 + 0.0789i -0.0004 - 0.0769i   0.0002 + 0.0283i
-0.1347 + 0.0000i   0.1533 + 0.0000i -0.0600 + 0.0000i
-0.3400 + 0.0000i   0.3602 + 0.0000i -0.1455 + 0.0000i
   0.5774 - 0.0000i -0.6800 - 0.0000i   0.2500 - 0.0000i
   0.3753 + 0.0000i -0.4700 - 0.0000i   0.1800 + 0.0000i
>>

=> The dimension of eigen space associaed with each eign value is the same as that of matrix A.

=> Since P is non-singular (det(P) not equal to zero), A is diagnolizable.

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