Linear Algebra Prove: If a, b, c, and d are integers such that a + b = c + d, th

Question

Linear Algebra

Prove: If a, b, c, and d are integers such that a + b = c + d, then has integer eigenvalues, namely xx = a + b and x2 = a - c. Prove: If A is a square matrix, then A and At have the same eigenvalues. Disprove by producing a counterexample: The eigenvalues of a matrix A are the same as the eigenvalues of the reduced row echelon form of A. Suppose that the characteristic polynomial of some matrix A is found to be p(lambda) = (lambda - 1)(lambda - 3)2(lambda -4)3. What can you say about the dimensions of the eigenspaces of A? What can you say about the dimensions of the eigenspaces if you know that A is diagonalizable? If {vi, v2, v3} is a linearly independent set of eigenvectors of A all of which correspond to the same eigenvalue of A, what can you say about that eigenvalue?

Explanation / Answer

5) | A - (lambda) * I | = 0

ad - ( b - lambda) ( c - lambda) = 0

ad = bc - lambda ( b + c) + lambda^2

lambda^2 - (b+c) lambda + ( bc - ad) = 0

Roots of the above equation are : 1) [ (b+c) + sqrt ( (b+c)^2 - 4(bc-ad)) ] / 2

= [ (b+c) + sqrt ( b^2 +c^2 + 2bc - 4bc + 4ad)) ] / 2

WE have a + b = c+ d, So b - c = d - a

= [ (b+c) + sqrt ( b^2 +c^2 - 2bc+ 4ad)) ] / 2

= [ (b+c) + sqrt ( ( b - c)^2 + 4ad)) ] / 2

= [ (b+c) + sqrt ( ( d - a)^2 + 4ad)) ] / 2

= [ (b+c) + sqrt ( (d+a)^2) ] / 2 = ((b+c) + ( d+a) )/ 2 = 2( a + b) / 2 = ( a+ b)

2) [ (b+c) - sqrt ( (b+c)^2 - 4(bc-ad)) ] / 2 = [( b+c) - ( a+ d) ] / 2 = a - c

Hence the roots are a+b and a - c

6) The matrix (A??I)T is the same as the matrix (AT??I), since the identity matric is symmetric.

Thus: det(AT??I)=det((A??I)T)=det(A??I)

From this it is obvious that the eigenvalues are the same for both A and AT.

7) Initial Matrix

After finding the following eigenvalues by finding the characteristic polynomial I get:

?1=?2=?2 and ?3=1

Taking matrix to echelon form:

multiplying row 1 by -0.2
adding -3 times row 1 to row 2
multiplying row 2 by 2.5
multiplying row 3 by -0.5

Taking matrix to reduced echelon form:

adding -1.2 times row 2 to row 1
adding -3 times row 3 to row 1
adding 3 times row 3 to row 2
Reduced echelon form:

Input matrix:

Eigenvalues Eigenvectors:

Eigenvalues:

Hence Proved.

-5 -6 3 3 4 -3 0 0 -2

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