In this problem yon will experiment with random matrices to try to determine whe

Question

In this problem yon will experiment with random matrices to try to determine whether each of the following "properties" of eigenvalues is either True or False: (a) The product of the eigenvalues of a matrix A Is equal to the determinate of A. (b) The eigenvalues of the power of a matrix equal the power of the eigenvalues of the matrix, i.e. if lambda is an eigenvalue of A, then lambda^k is an eigenvalue A^k. (c) Similar matrices have the same eigenvalues, i.e. if A = PBP^-1 then. A and B have the same eigenvalues. (d) Two matrices that are row equivalent have the. same eigenvalues. e.g. if B = rref (A) then A and B have the same eigenvalues. (e) The eigenvalues of a multiple of a matrix equal the same multiple of the eigenvalues of the matrix, i.e. if B = cA and if lambda is an eigenvalue of A, then c lambda is an eigenvalue of B. (f) The eigenvalues of the sum of two matrices are the sum of the eigenvalues of the two matrices, i.e. if lambda is an eigenvalue of. A and lambda is an eigenvalue of B, than (lambda + gamma) is an eigenvalue of (A + B). To draw conclusions about the above properties, I suggest that you use random 4 times 4 matrices; you can create these using MATLAB's rand function, e.g.

Explanation / Answer

a = rand (4,4)

a =

0.8147 0.6324 0.9575 0.9572
0.9058 0.0975 0.9649 0.4854
0.1270 0.2785 0.1576 0.8003
0.9134 0.5469 0.9706 0.1419

a) >> eig(a)

ans =

2.4021
-0.0346
-0.7158
-0.4400

prod (eig(a))

ans =

-0.0261

det (a)

ans =

-0.0261

hence true

b) true

eig(a^5)

ans =

79.9780
-0.1879
-0.0000
-0.0165

>> (eig(a)).^5

ans =

79.9780
-0.0000
-0.1879
-0.0165

c) true

p =rand (4,4)

p =

0.4218 0.6557 0.6787 0.6555
0.9157 0.0357 0.7577 0.1712
0.7922 0.8491 0.7431 0.7060
0.9595 0.9340 0.3922 0.0318

>> b =(inv (p))*a*p;
>> eig (b)

ans =

2.4021
-0.0346
-0.7158
-0.4400

as eigenvalues are same for matrix a and b

d) false

B = rref(a)

B =

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

>> eig(B)

ans =

1
1
1
1

eigenvalues are different

e) true

D = rand (1,1)

D =

0.2769

e = D*a;
>> eig(e)

ans =

0.6652
-0.0096
-0.1982
-0.1218

D*eig(a)

ans =

0.6652
-0.0096
-0.1982
-0.1218

f) false

f =rand (4,4);
>> eig(f+a)

ans =

4.3510 + 0.0000i
-0.6025 + 0.0000i
-0.0179 + 0.4757i
-0.0179 - 0.4757i

>> eig(f)+eig(a)

ans =

4.3668 + 0.0000i
-0.3631 + 0.0000i
-0.2834 + 0.3503i
-0.0076 - 0.3503i

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