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Hi guys I am reading the book for elementary linear algebra,

There is a question asking:

If $\lambda$ is a repeated eigen value then the corresponding eigen vectors are linearly depending.

But I am confused on the meaning here. If we have a repeated eigen value say multiplicity 2. Then we have an eigen vector and an generalized eigen vector that are independent. So in this sense this is false.

However the question said eigen vectors. But if we do not go to generalized eigen vectors we do get the same exact vector and multiples of it. Thus in this sense this is true.

Thus can people tell me how they are reading this, and comment. Thank you

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A repeated eigenvalue $\lambda$ of matrix $A$ with algebraic multiplicity $r$, i.e. a root of order $r$ of the characteristic polynomial, may have anywhere from $1$ to $r$ linearly independent eigenvectors. The number of linearly independent eigenvectors, i.e. the dimension of the null space of $A - \lambda I$, is called the geometric multiplicity of $\lambda$.

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I don't think the premise is correct if I'm understanding your English correctly. It doesn't follow that just because:

λ is a repeated eigenvalue then the corresponding eigenvectors are linearly depending.

I'm interpreting that as meaning that when an eigenvalue is repeated for a matrix that you're assuming that necessarily must correspond to eigenvectors that are linearly dependent. Which is not the case.

This StackExchange Answer solves a similar problem using parametric variables. You can test each of the eigenvectors in the example used on that page: they're linearly independent, yet are derived from a repeated eigenvalue.

Hope this helps.

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