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If I have a $2 \times 2$ matrix $A$, and then I find two eigenvalues $\lambda_1$ and $\lambda_2$ by subtracting $λI$ from $A$ and then taking the determinant=0(singular); to find $\lambda_1$ and $\lambda_2$.

So for a one eigenvalue $\lambda_1$, how many possibilities are there for eigenvectors? in another words, how many solutions are there?

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    There are at least as many solutions as there are nonzero elements in your field: if $\mathbf{v}$ is an eigenvector corresponding to $\lambda_1$, then so is $\alpha\mathbf{v}$ for every nonzero scalar $\alpha$.2012-02-27
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    For a given eigenvalue, the set of possible eigenvectors is a vector space (technically, a vector space minus $\{0\}$) called the eigenspace. So if you're working on real or complex vector spaces (or over any infinite field), there's an **infinite** number of possible eigenvectors. What might be a better measure of the size of the eigenspace is its dimension. We know it's at least $1$, and that the sum of the dimensions of all eigenspaces is at most the size of the matrix (in your case $2$).2012-02-27
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    so for eigenspace we can say that there is a whole line of eigenvectors?2012-02-27
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    @Binarylife: Not quite; the zero vector is not an eigenvector; and eigenspaces may have dimension greater than $1$, and so not be lines. But if $\lambda$ is an eigenvalue, then there is at least "a whole line, with the origin removed] of eigenvectors" corresponding to $\lambda$.2012-02-27
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    If you now understand the situation, Binarylife, you can post an answer yourself; if no one finds anything wrong with your answer, then you can accept it. This will give you valuable practice in writing things up.2012-02-28

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I hope you are asking for maximum possiblity of independent eigen vector: Below answer is for $2\times 2$ matrix.

We know that if $\lambda_1\neq\lambda_2$ then corresponding eigen vector will be independent. Below answer is based on this fact.

If $\lambda _1$ and $\lambda_2$ are different.... then there are only one independent eigen vector for corresponding eigen values.

If $\lambda_1$ and $\lambda_2$ are same then there may be two linear independent eigen vector.

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    So, If $\lambda_1$ = -1 and $\lambda_2$ = 2 , then I have only two corresponding eigenvectors which are {1,1} {5,2} , or there are other values maybe {2,2} instead of {1,1} and so on for the other eigenvalue?2012-02-27
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    @Binarylife , as other mention, if $v$ is eigen vector then $av$ will be eigen vector.. So for an eigen value, eigen vector is a vector space... Not just one... So if $(1,1)$ is eigen vector then $(2,2)= 2(1,1)$ will also be an eigen vector....2012-02-28