# Tagged Questions

1answer
39 views

### 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 $n \times n$ ...
1answer
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### Differential Equation: Complex Eigenvalue

For the following system $$x'=\left( \begin{array}{ccc} \frac{-1}{2} & 1 \\ -1 & \frac{-1}{2} \end{array} \right)x$$ To find a fundamental set of solutions, we assume that $$x = Ee^{rt}$$ ...
2answers
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### Eigenvalues and Eigenvectors of $X'X$ and $XX'$

I am trying to derive (or prove) the relationship between the eigenvalues and eigenvectors of the matrices $X'X$ and $XX'$. It is fairly intuitive that they are related but I cannot derive the ...
3answers
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### How to find solutions for linear recurrences using eigenvalues

Use eigenvalues to solve the system of linear recurrences $$y_{n+1} = 2y_n + 10z_n\\ z_{n+1} = 2y_n + 3z_n$$ where $y_0 = 0$ and $z_0 = 1$. I have absolutely no idea where to begin. I understand ...
2answers
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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 ...
2answers
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### Linear algebra, eigenvectors problem

Suppose you know that A is $2x2$ and symmetric. Assume the eigenvalues are $4$ and $7$. An eigenvector for $4$ is the vector $(3, -4)$. What is an eigenvector for $7$? So first we let ...
1answer
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### Linear algebra eigenvalues and limits problem

Suppose a matrix $A$ has eigenvalues of $-0.9$, $0.8$, $\pm 0.5$ and $0.9 \pm 0.2i$. What can you say about $\displaystyle\lim_{k \to \infty} A^k$ when $k$ approaches infinity ? Thanks.
1answer
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### by finding the eigenvalues and eigenvectors the evaluate the following.

so the question is : by finding the eigenvalues and eigenvectors of the matrix $$P=\begin{bmatrix}1&6\\0&-2\end{bmatrix}\qquad\text{evaluate }P^{20}\begin{bmatrix}-2\\1\end{bmatrix}$$ I ...
0answers
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1answer
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### What's the trick for proving one eigenvalue of orthogonal matrix is $-1$ if the determinant is $-1$?

Obviously, the magnitude of the orthogonal matrix is 1, which is easy to prove.. However, I wonder how can one prove that the eigenvalue of an orthogonal matrix is $-1$, if the determinant of this ...
3answers
63 views

### Finding eigenvectors with square root eigenvalues

I have a matrix $$\begin{bmatrix}1 &-1 &2\\2 &-2 &4\\0 &1 &1\end{bmatrix}$$ Its eigenvalues are $0$, $\sqrt{5}$ and $-\sqrt{5}$ (These are checked in MATLAB to be correct). I ...
0answers
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### Linear Algebra: Linear transformation and eigenvalues [duplicate]

Hi could some one please help. I am having problems proving this. Let $A$ be an $n \times n$ matrix with complex entries and let $f (t) =\det(A - tI)$ be its characteristic polynomial. Prove ...
0answers
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### Trouble with geometrical application of Lagrange multiplier

Let $A\in\mathbb R^{n\times n}$ be positive-definite and $\langle Ax,x\rangle=1$ be the equation of an ellipsoid $M\subset\mathbb R^n$. Use Lagrange multipliers to prove that the greatest distance of ...
1answer
42 views

### Hermitian matrices [duplicate]

Suppose we have a hermitian matrix $H$, and a matrix $A$ composed of eigenvectors of $H$, such that $\langle A_i \mid A_i \rangle =1$, where $A_i$ is the $i$-th column of matrix H. How to prove ...
2answers
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### eigenvectors and eigenvalues proof

I have deleted this question since there is a problem in the formatting
2answers
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### Are there analogues of eigenvalues/eigenvectors for a ring homomorphism/endomorphism?

My question is very simple. To put it in a context, a linear transformation is nothing but a homomorphism from a vector space to another. I usually visualize the action of a linear transformation by ...
1answer
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### Find eigenvalues and eigenvectors $f$ is $C^{\infty}$ $f \rightarrow f'$

Find eigenvalues and eigenvectors for a function $f$ which is $C^{\infty}$ and mapping $f \rightarrow f'$ which is an endomorphism. $\lambda$ is eigenvalue of a mapping $\varphi$ and $u$ is its ...
1answer
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### Adjugate matrix product

I have some problems understanding the proof of the Caley-Hamilton theorem (saying that a matrix the root of ith characteristic polynomial), namely: Why $A \cdot A^D = A^D \cdot A = \det A \cdot I$ ? ...
0answers
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### Hermitian matrix properties

Suppose we have hermitian matrix $H$, matrix $A$, composed of eigenvectors of $H$, such that $\langle A\mathbf i\mid A\mathbf i\rangle=1$, where $\mathbf i$ is the $i$-th column of matrix $H$. How ...
0answers
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### Delocalization of eigenvectors in Expanding Graphs

Given an adjacency matrix A, can we say something about whether the eigenvectors corresponding to its highest(or second-highest) eigenvalues are delocalized ? By delocalization I mean that every ...
2answers
63 views

### High-order elements of $SL_2(\mathbb{Z})$ have no real eigenvalues

Let $\gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbb{Z})$, $k$ the order of $\gamma$, i.e. $\gamma^k=1$ and $k=\min\{ l : \gamma^l = 1 \}$. I have to show that $\gamma$ ...
1answer
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### How to prove that the corresponding matrix is unitary

Let's say we are given hermitian matrix $H$. How to prove that the matrix $M$, formed from eigenvectors of $H$ is unitary? Thanks
1answer
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### Hermitian Matrices are Diagonalizable

I am trying to prove that Hermitian Matrices are diagonalizable. I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are ...
0answers
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### Probability Density Function and Eigenvalue Spectrum of Correlation Matrix

My question is in the link... http://www.flickr.com/photos/88684900@N03/8654322505/in/photostream
2answers
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### Dimension of the corresponding eigenspace?

I'm studying for my linear exam and would appreciate any help for this practise question: You are given that λ = 1 is an eigenvalue of A. What is the dimension of the corresponding eignspace? A = ...
1answer
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### Simultaneous eigenfunction problems

I'm familiar with solving eigenfunction problems using finite difference methods and eigenvalue solver like Eigensystem[] in Mathematica. But now I've come across a problem where I have two ...
1answer
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### Does this matrix have strictly positive eigenvalues?

Consider a square matrix $A$ with real entries and a diagonal matrix $B$ with strictly positive elements. Assume that the symmetric part of $A$, i.e. $\displaystyle \frac{A^T+A}{2}$, has strictly ...
0answers
35 views

### A Has characteristic polynomial that can be reduced to linear products $\Rightarrow$ A similar to upper triangular Matrix

Prove that if $A\in M_{n}\left(\mathbb{F}\right)$ matrix with a characteristic polynomial that can be written as a product of linear elements (?) ...
2answers
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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 ...
1answer
33 views