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Snehasish Ghosh posted an Question
June 25, 2020 • 16:46 pm
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30 points
30 points
- IIT JAM
- Physics (PH)
Why the two eigen vectors are different for the same eigen value.please explain
Why the two eigen vectors are different for the same eigen value.Please explain
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Chandra prakash
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it's a property that same eigen value doesn't mean same eigen vector.....every eigen value corresponds to a saperate eigen vector
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Chandra prakash
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The eigenvector for a given eigenvalue is not always unique. This is not always so, however, so when several eigenvectors have the same eigenvalue is is called a degenerate state.
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Dhairya sharma Best Answer
it's a property that same eigen value doesn't mean same eigen vector.....every eigen value corresponds to a saperate eigen vector
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here we have 2 arbitrary solution..... arbitrary solution = order - rank. here it's 2 so we got two different eigen vectors
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But how to know the value of the different eigen vectors when the eigen values are same?
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drar we check it from order - rank ....how many no we get these are no of eigen vectors
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and same eigen value doesn't mean same eigen vector. it's a property note it down
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this property asked too many times in exams ....becoz eigen vectors always defined uniquely
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Ghost.
First you need to understand that matrices are opertor and when matrices operate on any vector(coloumn matrices), it can do following operation. 1) Rotate coloumn Matrix in complex space. 2) Scale coloumn Matrix by some factor. Eigenkets of a matrices are those vectors , when get operated by operator they are only scaled(Multiplied) by a number(NOT ROTATED)and this number is called eigenvalue. for example (M)A=2*A (say) where M is a n*n matrix and A is a coloumn Matrix of order n*1. and 2 is a scaling factor. so there can be exist more than two column matrix which are scaled by same number without getting rotated,and this degeneracy is totally dependent on the nature of matrix. matrices(operator) can be frequently observe in Paul dirac notation(QUANTUM MECHANICS). the eigenkets of matrices are used as basis (just like x,y,z) to expand any vector, and to do so eigenkets must need to be linearly independent ( which is not the case in degeneracy,Neglecting GENERALIZED EIGENKETS) and span the whole space. REMEMBER Degeneracy comes at the cost of linearly independency. you can't span whole space with eigenkets of matrix which has degeneracy. Hope it helps.
Note even in the example you posted, eigenkets of degenerated value are not linearly independent. that is they are not orthogonal.