# Tagged Questions

For specific question about eigenvectors of a matrix or a linear operator.

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### Show that a transformation matrix is equal to the martix of eigenvectors

The real symmetric $3\times 3$ matrix $A$ has unit eigenvectors $\mathbf x_i$, $i=1,2,3$. Thus we have, $A\mathbf x_i=\lambda_i \mathbf x_i$. A $3\times 3$ matrix $C$ takes a vector in the ...
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### How to diagonalize a matrix

So I am trying to diagonalize this matrix {2,0,-2} {1,3,2} {0,0,3} so that those are the rows of the matrix. I know the eigen values are 2 and 3. I don't think that this matrix can be ...
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### Repeated eigenvalues: How to check if eigenvectors are linearly independent or not?

I have two related questions. My question is hypothetical, i.e. not from an actual physical problem. If I give you a matrix and tell you that it has a repeated eigenvalue, can you say anything about ...
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### Correspondence between eigenvalues and eigenvectors in ellipsoids

think of an ellipsoid in the n-dimensional space defined by $$(x-\mu)'A(x-\mu)=1.$$ I was calculating the volumes of n-dimensional ellipsoids like the one from above for a while, which is ...
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### Eigenspace and polynomials?

My prof introduced us to eigenvectors and eigenvalues today. He then gave us the following theorem: Theorem 6.6: Let $A$ be a square matrix, let $\gamma$ be an eigenvalue of $A$ with multiplicity ...
What is the relationship between eigenvalues of some square matrix $A$ and the eigenvalues of $A^k$ for some positive integer $k$? How about eigenvectors? I haven't touched linear algebra in a while ...