The Eigen-Values Of The Matrix

Running head: LINEAR ALGEBRA
LINEAR ALGEBRA
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1. Let the two matrixes A and B are
A =
[0 1 0 0
0 0 0 0
0 0 0 1
0 0 0 0 ], B =
[0 1 0 0
0 0 1 0
0 0 0 0
0 0 0 0 ]
Now, the eigenvalues of A are 0 and eigenvectors are
[1
0
0
0 ],
[−1
0
0
0 ],
[0
0
1
0 ],
[ 0
0
−1
0 ]
The eigenvalues of B are also 0 and so as the eigenvector.
As the eigenvalues and the eigenvectors are same hence the minimum polynomial equation,
Eigen space dimensions are also same.
Now, A2 = 0, B2 ≠ 0 and however, let B = P−1∗A∗P.
Then B2=P−1∗A2∗P
This becomes contradictory as if A2 is null matrix then P−1∗A2∗P needs to be null matrix.
Hence no matrix P exists such that B = P−1∗A∗P.
Hence, A and B matrixes are not similar but satisfies the same eigenvalues, eigenvector
conditions.
2. if two matrixes A and B are congruent over M4 then
B=M T∗A∗M
Then, det ( B )=det ( M T )∗det ( A )∗det ( M )
 det ( B)/det( A)=de t2 (M )(As det ( M T ) = det ( M ))
Hence, det (B)/det( A) must be a perfect square if the matrixes are congruent in M4.