# Tagged Questions

Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is and eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective.

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### Concerning the general solutions to linear ODEs Y'=AY when A has multiple eigenvalues

Given linear ODES Y'=AY, where Y is a column vector, A is a 6*6 square matrix. Clearly A has 6 eigenvalues, namely r1, r2, r3, r4, r5, r6. Herein we assume r5=r2, r6=r3.That is, r2 and r3 are two ...
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### Given the set of eigenvalues of a diagonalizable matrix, show that it satisfies an equation

Let $A$ be an $n \times n$ diagonalizable matrix. $A$ has only $2$ and $4$ as its eigenvalues. Show that $A^2 = 6A − 8I$. I get stuck on this question for a while. Can anyone give me a hint for this ...
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### Proving Invertibility and Eigenvalues

If matrix $A$ is an $n\times n$ matrix such that $A^2 -A -2I=0$. How can I show that $A$ is invertible and that $A^{-1} = \frac12(A-I)$? Also, how do i show that one of the eigenvalues of $A$ is 2 or ...
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### Diaonalized Matrix of the form $S^2=D$

If $D$ is a diagonal matrix, with non-negative eigenvalues, prove that there is a matrix $S$ such that $S^2 = D$
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### what would be the eigen vector for this value?

So I have a $3\times 3$ matrix $$A=\begin{pmatrix} 2&1&1\\ 1&2&1\\ 1&1&2 \end{pmatrix}.$$ My instructions are to find the eigenvalues and eigenvectors of the matrix. For each ...
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### B is A's Jordan form, so that there's an invertible matrix P so that $A=PBP^{-1}$. How do I find P?

B is A's Jordan form, so that there's an invertible matrix P so that $A=PBP^{-1}$. How do I find P? I tried solving, and here's a detailed path of my solution. Any help is very much appreciated! For ...
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### Prove the operators $T+U$ and $U$ have the same eigenvalues where $T$ is nilpotent

Let $V$ be an $n$-dimensional vector space on $\mathbb{C}$, and $T$ a nilpotent operator on $V$. Let $U$ be in $L(V)$ s.t. $UT = TU$. Prove that the operators $T+U$ and $U$ have the same eigenvalues. ...
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### Find formulas for the entires of $M^n$ ($n\ge0$) (Eigen Vector/Values)

I need a little help with an eigen vector question, The question Let $M =\left|\begin{matrix} 1 & 1 \\ -36 & 13 \end{matrix}\right|$ Find formulas for the entries ...
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### How to put $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ in canonical form

We are given the equation $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ We did an example of this in class but the equation had less terms. I took a note in class that says : if there are linear terms, I have ...
$A$ is a $7\times7$ invertible matrix , above the complex field. $\operatorname{rank}(A-I) = 3$, and $A^3 = A$ . I need to find the Jordan form of $A$. I'm stuck with some point: $A$ is invertible ...
Normally when one calculates the singular values for X these are the square roots of the matrix X'X. However here the X variables have been normalised so that $(x_{i,j}-\hat{x_j})/\sqrt{n-1}*s$ ...