Tagged Questions
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.
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23 views
maximum eigenvalue of a diagonal plus rank-one matrix
I want to compute the maximum eigenvalue of a diagonal plus rank-one matrix. Are there fast algorithms for the computation of the largest eigenvalue? What is the computational complexity of those ...
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0answers
27 views
Eigenvalues of differential operator
If $L : C^2[a,b] \rightarrow C^0[a,b] $ is s.t. $L y(t) = \ddot y(t) +p \dot y + q y(t) $ and $L$ is invertible then $L^{-1}$ has at most countable eigenvalues and they accumulate in $0$.
Why ...
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19 views
Eigenvectors for 2x2 upper triangle unit matrix
I will go straight to the problem. I have the following matrix:
[1 1] = A
[0 1]
When I determine the charecteristic polynomial, I obtain the eigenvalue 1 with multiplicity 2, i.e. lambda = lambda1 = ...
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0answers
20 views
Popov-Belevich-Hautus test for stabilizability
The system is: $x'(t) = Ax(t) + Bu(t)$
Which property do the uncontrollable eigenvalues of system need to have if system is
stabilizable, keeping the Popov-Belevitch-Hautus test for stabilizability ...
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votes
0answers
38 views
Diagonalizing using a matrix $P$
Let $A=\begin{pmatrix} a & b \\ c& d \end{pmatrix}$ be a $2 \times 2$ matrix witth eigenvalue $\lambda$.
(a) Show that unless it is zero, the vector $\begin{pmatrix} b \\ \lambda -a ...
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0answers
13 views
How is a the covariance matrix of a rotated dataset related its SVD and eigenvectors?
I understand how the covariance matrix can be used to find the orientation of a data cloud. For example, in 2-D, for zero mean data, the direction of the major axis of the cloud of data is given by ...
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votes
1answer
17 views
Recursive relation for a characteristic polynomial
I need to find a recursive relation for the characteristic polynomial of the $k \times k $ matrix $$\begin{pmatrix} 0 & 1 \\ 1 & 0 & 1 \\ \mbox{ } & 1 & . & . \\ \mbox{ } ...
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vote
1answer
26 views
Characteristic Polynomial of $A$ and polynomials annihilating $A$
If $A$ is a real $3 \times 3$ matrix which is not diagonal. $p$ is a polynomial of degree 3 with real coefficients which is annihilating $A$. I have proved that if $A$ has a complex root (with non ...
2
votes
1answer
75 views
Compute $e^{tA}$
When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(.
Here's my problem:
For the system of equations:
$$\begin{cases}
& \text{ } ...
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0answers
19 views
Compute $\|A(t)\|$
When I do my homework (stability theory), I must use the knowledge to the norm of matrix. But I don't remember it (I mean that I'm not sure).
My problem:
For matrix $A(t)$ is continuous. Compute ...
2
votes
3answers
157 views
Space of eigenvectors
I was wondering if there is any relation between the space span by all eigenvector of a matrix A and the column space of A. Also, is there any condition on A so that these two spaces are the same.
...
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1answer
35 views
Prove that if the Hessian of $f$ is positive definite at $a$, then the function attains a minimum at $a$
I'm trying to prove that if the Hessian $A$ of $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is positive definite at $a$, then the function attains a minimum at $a$.
I can't figure out how to use the ...
0
votes
1answer
26 views
about diagonal matrix and eigenvalues
I am reading the introduce of linear system and eigenvalues. There I read if there is a matrix $A$ and vector $x$, it could find a eigenvalue $\lambda$ such that $$Ax = \lambda x$$
I have a really ...
0
votes
1answer
37 views
Creating matrix from characteristic poynomial
How can we create a matrix from the characteristic polynomial? I know the procedure for creating the same when there are no repeated roots.
The procedure I used was to create a diagonal matrix with ...
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votes
0answers
25 views
Eigenvalues transformation
Consider $\mathbf{K}$ to be a $m \times m$ positive semidefinite matrix, which can be diagonalised as: $\mathbf{K} = \mathbf{P}_{m\times r} \boldsymbol{\Lambda}_{r \times r} \cdot ...
