For specific question about eigenvectors of a matrix or a linear operator.
0
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1answer
47 views
Intuitive meaning of right and left eigenvector
I am trying to get an intuitive understanding of the meanings of right and left eigenvectors. I guess the best thing you can do is to provide examples of application. (Examples from the field of ...
4
votes
1answer
40 views
Characteristic polynomial divides minimal polynomial if and only if all eigenspaces are one-dimensional
Prove that characteristic polynomial of a complex matrix $A$ divides its minimal polynomial if and only if all eigenspaces of $A$ are one-dimensional.
As far as I can see I the only possible case ...
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vote
1answer
40 views
Solving Linear ODE using matrices
What I don't understand here is where or how the operator for this solution is formed. Shouldn't the values of the operator be A=(1,0,0,1)? (in the form a11, a12, a13, a14 respectively).
Any help ...
4
votes
0answers
45 views
the spectrum and determinant of the Laplacian on $S^3$
I came across the following statement in a paper:
On $S^3$, the eigenvalues of the vector Laplacian on divergenceless vector
fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell ...
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votes
5answers
55 views
Eigenvalues and eigenvectors - help
I was reading my lecture notes on using matrices to solve ODEs and came across this and am having trouble understanding it:
(Please note that the two eigen values that are calculated are 2 and 0 ...
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votes
3answers
51 views
Finding the jordan form of a funny looking matrix
I was working on this problem from a previous qual exam. Just when I thought I knew how to find the Jordan form of any matrix and then I find this....aaaargh :)
Find the jordan form of the ...
3
votes
3answers
80 views
Why is $\det (A-\lambda I)=0$?
I'm not sure I understand the logic behind why $\det (A-\lambda I)=0$ for any non-trivial solution to $(A-\lambda I)x=0$.
1
vote
1answer
28 views
The generalized eigenvectors of linear operator $T$ span space $V$, why?
I'm studying about determinant and I have a problem understanding the following (Proposition 3.4):
The problems I have are highlighted with red rectangles. If anyone can, could you clarify these ...
2
votes
1answer
23 views
Relation between eigenvectors of covariance matrix and right Singular vectors of SVD, Diagonal matrix
I have a $m \times n$ data matrix $X$, ($m$ instances and $n$ features) on which I calculate the Covariance matrix $C$ and perform eigenvalue decomposition. so $C=W \Sigma W'$ where $W$ are the ...
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votes
0answers
14 views
recurrence relation basis vectors
I see that recurrence $x_n + c_1 x_{n-1} + \cdots = 0$, which may also be written as $x^n + c_1 x^{n-1} + \cdots = 0$ has a solution in the terms of eigenvalue powers, $$x_n = x_{01} \lambda_1^n + ...
5
votes
6answers
164 views
If $A\vec{v}=\lambda\vec{v}$, then does $A=\lambda$?
When my professor started teaching eigenvectors and eigenvalues the other day, the very first thing I noticed was the fact that $A\vec{v}=\lambda\vec{v}$ (assuming that the equation is satisfied under ...
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votes
0answers
11 views
low rank decomposition of composite PSD matrix
The matrix $M=AVA^T - BCB^T +D$ is known to be positive semidefinite (PSD), where $V, C, D$ are each diagonal matrices with positive values, and $V, C$ has small size when compared to the size of $M$. ...
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0answers
40 views
Has this Principal Component Analysis (PCA) been done correctly?
I have a set of 3D data points, indicated by the blue color in the picture below.
I then project them onto the x-y plane, i.e. setting z values of all the points to 0, shown by the yellow color ...
0
votes
0answers
24 views
Hermitian operator and its eigenbasis
Can anybody give me a simple proof for this theorem? the proofs that I have read are so confusing?
For every Hermitian operator $W$, there exists a basis consisting of its orthonormal eigenvectors ...
0
votes
1answer
24 views
Eigenvalues of a second derivative
I have a function f(r) that describes a Gaussian random field. A second derivative can be formed $\nabla_i \nabla_j f(r)$. I am looking at a paper that claims that in finding the extremum, the ...
