# Tagged Questions

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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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### If A is a square matrix, and $A^3$=I, how can I prove that eigenvalues of A are either 1 or -1?

If $A$ is a square matrix, and $A^3$=I, how can I prove that eigenvalues of $A$ are either $1$ or $-1$? Thanks for all your input, I figured it out! :)
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### Example of $5\times 5$ matrix with exactly $2$ distinct eigenvalues

What is an an example of a $5\times 5$ matrix with exactly $2$ distinct eigenvalues? Also, what would be the eigenvalues in the example?
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### Diagonalization, Solving a System of Linear Equations

I have questions regarding the following task, which is to diagonalize the matrix A: $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 9 & -20 \\ 0 & 4 & -9 \end{pmatrix}$ What I have ...
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### Solving for eigenvectors (Introduction to Linear Algebra by Serge Lang, example 2 page 240)

In "Introduction to Linear Algebra" by Serge Lang, example 2 page 240, there is the matrix $\begin{pmatrix}1&4\\2&3\end{pmatrix}$. The characteristic polynomial is $(t-5)(t+1)$. The system to ...
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### A quesion in Fulton & Harris book “representation theory a first course”

In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says "If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then ...
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### complex eigenvectors with non zero real parts

I'm wondering about how to deal with complex numbers in eigenvectors that have non zero real parts, as in my eigenvector is $\bigl[\begin{smallmatrix}1-2i\\-1\end{smallmatrix}\bigr]$ that is supposed ...
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### how do I prove that two matrices with same determinant and trace have different eigenvalues?

Assuming that they are both Hermetian, positive definite and have the same full rank. (To show the converse that if two matrices have the same eigenvalues, they must have the same determinant is ...
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### Bounding the smallest eigenvalue of an ergodic Markov Chain

I am trying to prove that the smallest eigenvalue of an ergodic Markov chain is greater than -1. Can we do that using proof by contradiction, i.e. assuming the smallest eigenvalue being -1, etc.? The ...
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### An eigenvector is a non-zero vector such that…

Various sources define eigenvalues and eigenvectors in slightly different ways (context independent). For example, both of the following definitions seem not to exclude the zero-vector as an ...
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### Condition number and Chebyshev systems

Suppose I have a square matrix $A$ of size $n$ with elements $a_{mn}=\phi_m(x_n)$ where $\phi_m(x)$ can be thought of as a very friendly function: orthogonal, bandlimited, bounded and analytic. Also, ...
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### Eigenvalues of $A$

Let $A$ be a $3*3$ matrix with real entries. If $A$ commutes with all $3*3$ matrices with real entries, then how many number of distinct real eigenvalues exist for $A$? please give some hint. ...
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### Property of sequence of eigenvalues of an operator.

For a positive (self adjoint) operator $A$ with eigenvalues $\lambda_k$, is it possible to have the case when neither $\lambda_k\to \infty$ or $sup_k \lambda_k<\infty$ for example if a subsequence ...
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### Evaluating eigenvalues of a product of two positive definite matrices

Let $A,B\in M_n(\mathbb{R})$ be two symmetric positive definite matrices, i.e.: $$\forall x\in\mathbb{R}^n, x\neq 0, (Ax,x)>0, (Bx,x)>0,$$ where $(\cdot,\cdot)$ is the usual scalar product in ...
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### A basic example of basis for eigenspaces

Hi I'm just learning how to find bases for eigenspaces and I ran into a very basic case that confused me. So the matrix is $$A = \begin{pmatrix}2 & 1\\0 & 2\end{pmatrix}$$ and $\lambda = 2$. ...
Assume I have some constant matrix $A$ to which I add a perturbation, resulting in $M(\epsilon )=A+\epsilon B$ the perturbed matrix ($B$ is constant as well), and that I can easily find the ...
### How to compute $\text{trace}((A+D)^{-1}A)$
Give a diagonal perturbation matrix $D$ (which is not an identity matrix), is there a simple way to compute $$\text{trace}((A+D)^{-1}A)$$ Or is there a good approximation?