For specific question about eigenvectors of a matrix or a linear operator.
1
vote
1answer
41 views
Fast way to calculate Eigen of 2x2 matrix using a formula
I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html
Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
1
vote
3answers
34 views
Eigenvector Proof $(I+A)^{-1}$.
Show that the eigenvectors of the $n \times n$ matrix A are also eigenvectors of the matrix $$M = (I+A)^{-1} $$ Where I is the $n \times n$ unit matrix. Determine the eigenvalues.
My Work:
...
0
votes
0answers
21 views
Gershgorin interval of an eigenvalue and the largest coordinate of the corresponding eigenvector
Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries.
The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has all coordinates, ...
1
vote
1answer
38 views
Solve a System with Variable
Given these matrices, how does one find two real solutions?
$dx/dt$ =
$\begin{bmatrix}
3 & -5\\
5 & 3
\end{bmatrix}x$
with $x(0) = \begin{bmatrix}
2\\
-3
\end{bmatrix}$
1
vote
1answer
34 views
Same eigenvalues, different eigenvectors but orthogonal
I am using a two different computational libraries to calculate the eigenvectors and eigenvalues of a symmetric matrix. The results show that the eigenvalues calculated with both libraries are exactly ...
0
votes
2answers
49 views
Common eigenvector of two linear transformation matrices
I have two linear transformation matrices
\begin{pmatrix}
3 & 2 \\
-2 & 1
\end{pmatrix}
and
\begin{pmatrix}
1-a & -a \\
a & 1
\end{pmatrix}
How to find out what the value of ...
1
vote
1answer
40 views
generalized eigenvector for 3x3 matrix with 1 eigenvalue, 2 eigenvectors
I am trying to find a generalized eigenvector in this problem. (I understand the general theory goes much deeper, but we are only responsible for a limited number of cases.)
I have found ...
3
votes
2answers
72 views
Diagonalising a $2 \times 2$ and $3 \times 3$ matrix
For each of the following matrices $A$, find an invertible matrix $P$ over $C$ such that
$P^{-1}AP$ is upper triangular:
$$A = \begin{bmatrix}4 & 1\\-1 & 2\end{bmatrix} \quad \text{ ...
0
votes
1answer
33 views
Normal matrices with orthogonal basis
we have a theorem that says that each REAL normal matrix can be written in terms of an orthonormal basis, so that it has its eigenvalues down the diagonal and 2x2 matrices of the form $\begin{pmatrix} ...
1
vote
2answers
54 views
Finding the eigenvectors of a matrix.
Find the normalized eigenvectors of this matrix:
$A=\begin{bmatrix}2&1&1\\1&2&1\\1&1&2\end{bmatrix}$
My work:
$$\det(A-\lambda I) =(\lambda-1)^2(\lambda-4)$$
$$\lambda_1=1, ...
2
votes
1answer
49 views
Linear Algebra - Can I use RREF to solve augmented matrix for eigenvector?
I am solving David Lay's 4th edition 7.1 number 16. So here's the problem. The original matrix: $$\begin{bmatrix}-7 & 24 \\ 24 & 7\end{bmatrix} \ ,$$ with eigenvalues $\pm 25$.
I am having ...
4
votes
0answers
54 views
+50
Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?
An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
1
vote
0answers
23 views
Deriving left eigenvector from the left eigenvector of transpose
How would I find the left eigenvector of A * A' if the left eigenvector of of A' * A is v?
4
votes
1answer
75 views
Interpretation of eigenvectors of cross product
If we fix a non-zero vector $\boldsymbol{v}\in\mathbb{R}^3$, then the linear map $\boldsymbol{x}\mapsto\boldsymbol{v}\times\boldsymbol{x}$ has trivial eigenvectors $\boldsymbol{x}_1=t\boldsymbol{v}$ ...
0
votes
0answers
32 views
nodal lines in the dirichlet problem on unit disk
In the Dirichlet problem if nodal lines do not touch $\partial\Omega$ (unit disk), what happens to the eigenvalues?
Thanks for help
1
vote
0answers
45 views
Vandermonde question
I'm studying time series analysis and in my book I came a cross with the following proof (The proof is actually the last page, but I posted as much information as possible on the problem):
I have ...
0
votes
2answers
34 views
Does a constant eigenvalue of a linearly parameter-dependent matrix have a constant eigenvector?
Let $A(\alpha)=A_0+\alpha A_1$, with $A_0,A_1\in\mathbb R^{n\times n}$ and $\alpha\in\mathbb R$, such that there exists a $\lambda\in\mathbb C$ with the property that
for all $\alpha$: $\det(\lambda ...
1
vote
1answer
36 views
Eigenpairs Differential equation
Consider the linear system Y'=AY given by Y'=\left[\begin{matrix}3 & 2 \\ -9 & -6\end{matrix}\right]Y
A. Compute the eigenpairs for the coefficient matrix A
So far I have lambdas= 0 and -3 ...
2
votes
0answers
32 views
MATLAB eig(A,B) of a positive semi-definite matrix pair (A,B) gives complex eigenvalues
A and B both are 151 by 151 psd matrices, I want to solve the generalized eigenvalue proplem
A * v = Lambda * B * v
when using MATLAB function eig()
[V,D] = eig(A,B)
I get complex numbers in both ...
1
vote
1answer
54 views
problem with 4x4 matrix with big elements
I have a homework for my linear algebra class at my university the thing is that we get a 4x4 matrix A then we have to find it's Transpose which is pretty easy and then find the matrix B=(A^T)*A also ...
0
votes
1answer
63 views
Eigenvector with eigen value of 1
How is an eigenvector with eigen value of 1, say v, multiplied by its transpose the identity matrix? v' * v = I?
-1
votes
1answer
25 views
diagonalizing a matrix $A$: can $P$ be bigger than $A$?
can you have a P bigger than the original A matrix? in other words after I found the eigenvalues I then found all the eigenvectors so when I constructed the P vector turns out to be bigger than my ...
1
vote
2answers
58 views
Eigenvector of matrix of equal numbers
For matrix the matrix
$$A = \begin{bmatrix}
3&1&1\\
1&3&1\\
1&1&3\\
\end{bmatrix}$$
with eigenvalues $\lambda_1=5$, $\lambda_2=2$, $\lambda_3=2$, I am trying to find the ...
0
votes
0answers
32 views
Diagonalizing the sum of a matrix and a multiple of the identity matrix
Suppose we have a matrix $A = B+\lambda I$, where $B\in \mathbb{R}^{n\times n}$, $I$ is the identity matrix and $\lambda\in \mathbb{R}$. If I know the eigenvalues and eigenvectors of $B$, what can I ...
0
votes
2answers
25 views
Matrix multiplication and eigen vectors
If $a$ is a right eigenvector of $S$ and $R^T$ with eigenvalue $1$. How would determine $a^TRSa$? Is $Sa$ simply $a$? Any hints that apply here would be greatly appreciated.
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
votes
0answers
32 views
Collapsing of data points without Normalisation [closed]
Am trying to formulate a model with different data sets. What is the best way to bring them together without using normalisation? What algorithms?
2
votes
2answers
58 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
0answers
20 views
Simultaneous orthogonalization for rectangular matrices?
Say that I have two linearly independent vectors in 3 dimensions, $\vec{a}_1$ and $\vec{b}_2$:
$$\vec{a} = \left[a_{1},\ a_{2},\ a_{3}\right]$$
$$\vec{b} = \left[b_{1},\ b_{2},\ b_{3}\right]$$
Using ...
1
vote
1answer
38 views
Finding two eigenvectors from one eigenvalue
My assignment tells me to Find a diagonal matrix similar to the matrix: $$\begin{bmatrix}0 &-4 &-6\\-1 &0 &-3\\1 &2 &5\end{bmatrix}$$ It's eigenvalues are 1, 2 and 2. And the ...
1
vote
0answers
22 views
network centrality: when does the most central node coincide for eigenvalue and degree centrality measures
I'm trying to understand when the most central node of a graph is the same when measured according to the degree- and eigenvalue-centrality measures. That is, if $\mathbf{x}$ is the principal ...
3
votes
2answers
46 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 ...
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} ...
2
votes
3answers
62 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 ...
0
votes
5answers
55 views
Having Difficulty Finding Eigenvectors
I'm having a lot of problems with the following problem in Steven J. Leon's "Linear Algebra with Applications" 8th edition. Problem 6.1.1.I asks the reader to "find the eigenvalues and the ...
0
votes
1answer
77 views
Finding the Matrix Power of a matrix and limit
Find the matrix power, $A^k$, of
$$A=\begin{pmatrix}a & 1-a \\ b & 1-b\end{pmatrix}$$
$$D=P^{-1}AP$$
$$A^k=PD^kP^{-1}$$
I think that
$$P=\begin{pmatrix}1 & \frac{a-1}{b} \\ 1 & ...
0
votes
1answer
42 views
Hermitian matrices [duplicate]
Suppose we have a hermitian matrix $H$, and a matrix $A$ composed of eigenvectors of $H$, such
that
$\langle A_i \mid A_i \rangle =1$, where $A_i$ is the $i$-th column of matrix H.
How to prove ...
0
votes
2answers
56 views
eigenvectors and eigenvalues proof
I have deleted this question since there is a problem in the formatting
2
votes
2answers
48 views
Are there analogues of eigenvalues/eigenvectors for a ring homomorphism/endomorphism?
My question is very simple. To put it in a context, a linear transformation is nothing but a homomorphism from a vector space to another. I usually visualize the action of a linear transformation by ...
0
votes
0answers
48 views
Hermitian matrix properties
Suppose we have hermitian matrix $H$, matrix $A$, composed of eigenvectors of $H$, such that $\langle A\mathbf i\mid A\mathbf i\rangle=1$, where $\mathbf i$ is the $i$-th column of matrix $H$.
How ...
1
vote
3answers
31 views
Calculate eigenvectors
I am given the $2\times2$ matrix $$A = \begin {bmatrix} -2&-1 \\\\ 15&6 \ \end{bmatrix}$$
I calculated the Eigenvalues to be 3 and 1. How do I find the vectors? If I plug the value back into ...
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
1answer
24 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
100 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 ...
2
votes
2answers
54 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
0answers
33 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 ...
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!
0
votes
1answer
47 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.
