28 views can someone please explain to me how to find the no of independent eigenvectors?

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For finding eigen vectors, we use the characteristic equation which is a homogeneous system of linear equations

[ A-kI ]X =0

where k is eigen value and I is identity9 matrix and X is eigen vector corresponding to eigen value k.

For given matrix which is a identity matrix of order 2, eigen values are 1,1 ie repeated igen values. Here k=1 only.

First find the matrix [ A-kI ] and find the rank of matrix. Let rank of matrix be r.

Here value of r is 0 as [A-kI] will be null matrix of order 2.

Then number of linearly independent eigen vectors= number of linearly independent solutions of homogeneous set of equations= n-r
where n is the order of given matrix
Here number of LI eigen vectors= 2-0=2
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Or it can also be solved as follow,

We know that this is identity matrix and upon multiply any vector with this matrix vector will not get changed.

Now we also know that eigen vectors of a given matrix are vectors which only gets scaled(either negetively or positively) after matrix multipplication.

So here every vector from 2D space you pick is eigen vector of this matrix. And there are only 2 linearly independant vectors in this space. so answer is 2.
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Thank you. Nicely explained ✌️

Bt what if there were 2 distinct Eigen values ??
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Here is more general statement: If you have all distinct eigen values then that matrix has all eigen vectors linearly independent(means n independent eigen vectors for nxn matrix.). But converse is not always true.