how to find matrix of eigenvector and eigenvalues are given
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Step-by-step explanation:
Writing the matrix down in the basis defined by the eigenvectors is trivial. It's just
M=⎛⎝⎜1000−20002⎞⎠⎟.
Now, all we need is the change of basis matrix to change to the standard coordinate basis, namely:
S=⎛⎝⎜10−1111−12−1⎞⎠⎟.
This is just the matrix whose columns are the eigenvectors. We can change to the standard coordinate bases by computing SMS−1. We get
SMS−1=16⎛⎝⎜1−8−5−84−8−5−81⎞⎠⎟.
You can check that this matrix has the desired eigensystem. For example,
16⎛⎝⎜1−8−5−84−8−5−81⎞⎠⎟⎛⎝⎜−12−1⎞⎠⎟=⎛⎝⎜−24−2⎞⎠⎟.
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Jan 26 '15 at 0:18
Mark McClure
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Jan 26 '15 at 1:28
sorry, I don't get it. Shouldn't be the result Ax = λx for all the three pairs of values and vectors? Like for example for the first pair: result-matrix * u1 = 1 * u1? – user3435407 Jan 26 '15 at 0:54
@user3435407 Yep - I guess I don't get what you don't get. I added one check. – Mark McClure Jan 26 '15 at 1:28
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People: If you were searching for the answer of this question too, now I found a good live demonstration of the above mentioned solution here: youtube.com/watch?v=HWnCv4iHCDc – user3435407 Jan 26 '15 at 3:40
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call the eigenvectors u1,u2 and u3 the eigenvectors corresponding to the eigenvalues 1,−2, and 2. then
A=1u1uT1uT1u1−2u2uT2uT2u2+2u3uT3uT3u3
you can verify this by computing Au1,⋯. this expression for A is called the spectral decomposition of a symmetric matrix.