Number 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.
0
votes
0answers
21 views
Properties of eigenvectors
If a is a right eigenvector of Z and b is a right eigen vector of Y, is a * b' a right eigenvector of Z * Y?
1
vote
2answers
27 views
Computing eigenvalues for $\mathrm{Sym}^2(\mathrm{Sym}^3 V))$ for $V = \Bbb C^2$
Given $V = \Bbb C^2$ the standard representation of $\mathfrak{sl}_2\Bbb C$, on page 157 of Fulton and Harris's Representation Theory they state
Since $U = \mathrm{Sym}^3 V$ has eigenvalues $-3, ...
4
votes
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 $n \times n$ ...
1
vote
1answer
48 views
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 $$ x = Ee^{rt}$$
...
2
votes
2answers
60 views
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 ...
0
votes
3answers
31 views
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 ...
3
votes
2answers
55 views
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 ...
3
votes
2answers
47 views
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 ...
0
votes
1answer
23 views
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.
-3
votes
1answer
42 views
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 ...
1
vote
0answers
40 views
Help find a proof : $ \lambda $ is $f$'s eigenvalue then $f|_{V_{\lambda}} $ has Jordan's basis
Could you help me find a fairly simple proof of the following theorem?
$f: V \rightarrow V, \ \ \dim V < \infty, \ \ \lambda$ is $f$'s eigenvalue $\Rightarrow \ \ f|_{V_{\lambda}}: \ V_{\lambda} ...
0
votes
1answer
19 views
Let $0 \ne u \in \mathbb{C}^n$ fixed and consider for every $v \in\mathbb{C}^n$ the matrix $E(v)=uv^*$. Give a polynomial $p$ such that $p(E)=0$.
Let $0 \ne u \in \mathbb{C}^n$ fixed and consider for every $v \in
\mathbb{C}^n$ the matrix $E(v)=uv^*$.
Give a quadratic polynomial $p$ such that $p(E)=0$.
I know that the eigenvalue of ...
0
votes
1answer
24 views
Let $0 \ne u \in \mathbb{C}^n$ fixed and consider for every $v \in \mathbb{C}^n$ the matrix $E(v)=uv^*$. Find all eigenvalues of $E(v)$.
Let $0 \ne u \in \mathbb{C}^n$ fixed and consider for every $v \in
\mathbb{C}^n$ the matrix $E(v)=uv^*$.
Find all eigenvalues of $E(v)$.
$E(v)=uv^*=
\begin{pmatrix}
u_1\\
\vdots \\
u_n
...
2
votes
1answer
105 views
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 ...
2
votes
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 ...
1
vote
0answers
44 views
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 ...
0
votes
0answers
23 views
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 ellipsoid $M\subset\mathbb R^n$. Use Lagrange multipliers to prove that the greatest distance of ...
0
votes
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 ...
0
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2answers
57 views
eigenvectors and eigenvalues proof
I have deleted this question since there is a problem in the formatting
2
votes
2answers
48 views
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 ...
0
votes
1answer
18 views
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 ...
1
vote
1answer
39 views
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 $A \cdot A^D = A^D \cdot A = \det A \cdot I$ ?
...
0
votes
0answers
49 views
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 ...
1
vote
0answers
18 views
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 ...
1
vote
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 $\gamma$ ...
1
vote
1answer
25 views
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
1
vote
1answer
107 views
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 ...
0
votes
0answers
25 views
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
2
votes
2answers
57 views
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 = ...
0
votes
1answer
22 views
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 ...
3
votes
1answer
70 views
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 ...
0
votes
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 (?) ...
2
votes
2answers
37 views
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 ...
3
votes
1answer
33 views
Convergence of eigenvalues $\lambda_{i}^k$
I've been trying to solve this for a while now but can't seem to figure it out. My initial intuition is to just calculate the limit of $\lambda_{2}^k$ using de Moivre's formula like so:
$\lim_{k\to ...
5
votes
1answer
140 views
Condition of the eigenvalue problem
[Ciarlet, 2.3-1] I know this result:
Let $A$ a diagonalisable matrix, $P$ a matrix such that
$$P^{-1}AP\ =\ \mbox{diag}(\lambda_i)\ =\ D,$$
and $\|\cdot\|$ a matrix norm satisfying
...
3
votes
2answers
67 views
A linear algebra problem
I came across the above problem .Everything looks fine until the last line that says $\longrightarrow$It yields the solution $y_1=(e^t, 0)^t$. I do not know how it came into the picture. A ...
0
votes
1answer
24 views
Lower bound for divergence of matrix spectrum
We consider a matrix $D$ of size $N \times N, N > 1$.
Denote by $\lambda_1 \geq \ldots \geq \lambda_N$ the eigenvalues of this matrix.
The goal is to provide a lower bound for the quantity ...
2
votes
1answer
41 views
Finding $A^n$ in terms of $P$ and $D$ (diagonalized)
My question is regarding the last two parts. I have Found $D$ and $P$, how can I obtain $A^{200}$ and det $(A^{200})$ form $PDP^{-1}?$ Thanks!
4
votes
1answer
63 views
Importance of eigenvalues
I know how to find eigenvalues and eigenvector .But I dont know what to do with that.
What is there use? Can anyone explain me that?
0
votes
1answer
48 views
Finding eigenvalues and eigenvectors
Find the eigenvalues and eigenvectors for the matrix A = (1,2,0), (-1,-2,1), (2,4,1). I end up with the polynomial x^3-5x=0 and no exact solution.
2
votes
1answer
50 views
Eigenvectors and Principal component
What is the difference between eigenvectrors and principal component. I got confused about this point because some researches reported that the principal components are the same eigenvectors of ...
1
vote
2answers
47 views
Eigenvectors and Kernel of Matrix
I'm trying to take find the eigenvectors of the matrix
$$
\begin{bmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{bmatrix}.
$$
I've found the eigenvalues of $1$ and $0$. I'm ...
2
votes
1answer
29 views
How do I generate my linear transformation matrix in this example?
Let $T$ be the linear transformation of space of polynomials $P^3$ given by
$$T(a + bx + cx^2) = a + b(x + 1) + c(x + 1)^2$$
Find all eigenvalues and eigenvectors of $T$.
4
votes
1answer
66 views
A matrix eigenvalue question
If $A, B, C$ are positive definite matrices of size $n$, is it true that $\lambda_j(A(B+C)^2A)\ge \lambda_j(AB^2A)$, $j=1, \dots, n$? $\lambda_j$ means the $j$-th largest eigenvalue.
1
vote
0answers
20 views
Latent semantics and eigenvalues of the SVD
Maybe a strange question :-) trying to figure out the relation with eigenvalues and the documents/words of and LSI? since $Av = \lambda v$, does this mean that eigenvalues describe the relation ...
1
vote
1answer
95 views
A question on Eigenvalues and Eigenvectors
Hey I need quite a bit of help, I know the question might seem easy, but I'm confused with the wording.
I know that the rule is $Av = \lambda(v)$.
Since there are two eigenvalues, I am assuming ...
2
votes
3answers
71 views
Linear Algebra Question on Eigenvalues
I am having a difficult time with the following question. Any help will be much appreciated.
Let $A$ be an $n×n$ real matrix such that $A^T = A$. We call such matrices “symmetric.” Prove that the ...
1
vote
1answer
78 views
If $A$ is a symmetric matrix, show that every eigenvalue of $A$ is nonnegative if and only if $A=B^2$
If $A$ is a symmetric matrix, show that every eigenvalue of $A$ is nonnegative if and only if $A=B^2$ for some symmetric matrix $B$.
My idea was to make use of the fact that $A$ is symmetric and thus ...
1
vote
2answers
56 views
Adjacency, Laplacian and Maximum Degree
I am relevantly new to http://math.stackexchange.com/ and this will be my first question, although I have been lurking around for some time now! So, pardon me if I am missing any posting etiquette ...
1
vote
0answers
23 views
finding the decomposition of Laplacian matrix with position of zero elements unchanged
I'd like to know whether it's possible to find the decomposition of a Lapalacian matrix $A$
$B^TB = A$
where $B$ has the same dimension with $A$ and the position of zero elements in $B$ is the same ...
