# Tagged Questions

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### To prove that the dimension of $V$ is $d_1^2 + \ldots + d_k^2$

Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $$(x - c_1)^{d_1} \cdots (x - c_k)^{d_k} ,$$ where $c_1,\ldots,c_k$ are distinct. Let $V$ be the space of $n \times n$ ...
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### Proof of the linear independence of the generalized eigenvectors of a square matrix

I'm currently stuck on this problem: Let $V$ be a finite dimensional vector space. If $S: V\rightarrow V$ and $T: V\rightarrow V$ are linear maps and $ST=TS$, prove every eigenvalue of $ST$ is a ...
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### Generalised eigenvalue is eigenvalue if it is in the field

I would like to prove the following assertion: Let $\mathscr{F}$ be a field and $\mathscr{\phi}$ be an $\mathscr{F}$-linear endomorphism of a finite dimensional $\mathscr{F}$-vector space ...
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### Eigenvalues and Eigenvectors Diagonilization

Let $A=\begin{bmatrix} -7 & -1 \\ 12 & 0 \\ \end{bmatrix}$ . Find a matrix $P$ and a diagonal matrix $D$ such that $PDP^{-1} = A$. Ok so the first thing I need to look ...
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### Finding point distribution by eigen vectors

First of all I want to tell that my mathematics is poor, so I canâ€™t use correct terms. Sorry for that. I have a point data set. This data represents some cylindrical objects surfaces (not exactly ...
### $2\times 2$ matrices over $\mathbb{C}$ that satisfy $A^3=A$
Let $A$ be a $2\times 2$ matrix with complex entries. What would be the number of $2\times 2$ matrices $A$ that satisfies $A^{3} = A$. Question was are they infinite? If it is $3\times 3$ matrix then ...
Suppose a square, real and symmetric matrix $G\in\mathbb{R}^{n\times n}$ is given, and it is known to have one zero eigenvalue associated with all ones eingenvector, $1_n$. I'm aware that the ...