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Snehasish Ghosh posted an Question

June 27, 2020 • 02:20 am

30 points

IIT JAM
Physics (PH)

How to know whether a matrix is diagonalizable or not? please explain

How to know whether a matrix is diagonalizable or not? Please explain

3 Answer(s)

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Aditi agrawal

how we decide eigen vectors for same eigen values
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means there will be one eigen vector for one eigen value or more..??
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also in the example posted by snehasish ghosh the two eigen vectors for one eigen value are also not orthogonal so how to decide how many eigen vectors can be taken
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For same eigen values if the matrix is non symmetric, then we have to find the conditions by putting the values for lambda and for symmetric matrices with repeated eigen values this orthogonal condition comes into effect
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which conditions we have to find and i am also asking for non symmetric matrices
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Ghost.

First of all you have to understand every matrix is written in centain basis(x,y,z) and another important thing is eigenvalue are only observable which a matrix can provide, a n*n matrix is a n-dimensional matrix, when you diagonalize a matrix, you look for certain basis in which structrual form of matrix can be simplified and observable can be easily visible. now difficult work is to find this set of vectors which can serve as basis, which we get as eigenkets of a matrix, and value associated with eigenkets is eigenvalue, now if your matrix has no degeneracy and non singular than you can easily find the basis and diagonalize matrix, but if you matrix has degeneracy involved then it may or may not be possible to find this basis. therefore in order to diagonalize a matrix you need to find n linearly independent set of eigenvectors, and it completely depend upon the nature of your matrix. Wishes.
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M
Mahak

diagonalization is possible only when eigenvalues are non degenrate for eg if we have a 3*3 matrix and this matrix have 3 linearly independent eigenvalues and eigenvectors then diagonalization is possible
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degenerate means?
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degenrate means when a n*n matrix have more than one same eigenvalue and non degenrate means when a n*n matrix have all the n eigenvalues different
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Ok Ma'am, Thank you
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Chandra prakash

see attached file its method and some example... I hope it will help uh dear..
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see attached file
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Got is sir...
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Dhairya sharma

if you have less than n linearly independent eigen vectors for n*n matrix then determinant will be zero and P(inverse) will not be possible...
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see this
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Snehasish ghosh

this
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Snehasish ghosh

see this
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M
Mahak

see this also
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Ma'am, but if two eigen values are equal,then too we have a diagonal matrix
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in that case eigenvectors may be linearly independent of each other
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How?
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means the two eigenvectors does not depend on each other
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Ok Ma'am
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Ruby negi Best Answer

if you have less than n linearly independent eigen vectors for n*n matrix then determinant will be zero and P(inverse) will not be possible...
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Ok Ma'am
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Ruby negi

if n*n matrix has n linearly independent Eigen vectors then that matrix is diagonalizable...
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