Tagged Questions
1
vote
1answer
46 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
23 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
39 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 $x(0) = \begin{bmatrix}
2\\
-3
\end{bmatrix}$
0
votes
2answers
50 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
2answers
55 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:
$$\det(A-\lambda I) =(\lambda-1)^2(\lambda-4)$$
$$\lambda_1=1, ...
1
vote
0answers
46 views
Vandermonde question
I'm studying time series analysis and in my book I came a cross with the following proof (The proof is actually the last page, but I posted as much information as possible on the problem):
I have ...
1
vote
1answer
57 views
problem with 4x4 matrix with big elements
I have a homework for my linear algebra class at my university the thing is that we get a 4x4 matrix A then we have to find it's Transpose which is pretty easy and then find the matrix B=(A^T)*A also ...
0
votes
1answer
63 views
Eigenvector with eigen value of 1
How is an eigenvector with eigen value of 1, say v, multiplied by its transpose the identity matrix? v' * v = I?
1
vote
2answers
59 views
Eigenvector of matrix of equal numbers
For matrix the matrix
$$A = \begin{bmatrix}
3&1&1\\
1&3&1\\
1&1&3\\
\end{bmatrix}$$
with eigenvalues $\lambda_1=5$, $\lambda_2=2$, $\lambda_3=2$, I am trying to find the ...
0
votes
0answers
33 views
Diagonalizing the sum of a matrix and a multiple of the identity matrix
Suppose we have a matrix $A = B+\lambda I$, where $B\in \mathbb{R}^{n\times n}$, $I$ is the identity matrix and $\lambda\in \mathbb{R}$. If I know the eigenvalues and eigenvectors of $B$, what can I ...
3
votes
2answers
46 views
Linear algebra, eigenvectors problem
Suppose you know that A is $2x2$ and symmetric.
Assume the eigenvalues are $4$ and $7$.
An eigenvector for $4$ is the vector $(3, -4)$. What is an eigenvector for $7$?
So first we let ...
0
votes
5answers
55 views
Having Difficulty Finding Eigenvectors
I'm having a lot of problems with the following problem in Steven J. Leon's "Linear Algebra with Applications" 8th edition. Problem 6.1.1.I asks the reader to "find the eigenvalues and the ...
0
votes
1answer
83 views
Finding the Matrix Power of a matrix and limit
Find the matrix power, $A^k$, of
$$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
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
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
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
0answers
34 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
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!
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
2answers
30 views
The connection between the rank of a matrix and its zero-mode eigenvectors
I would be most thankful if you could help me prove that if an arbitrary n by n matrix has rank m < n, then the matrix has (n-m) linearly independent eigenvectors corresponding to the eigenvalue ...
0
votes
0answers
35 views
Limit of $A^k b$ for $k$ large
My textbook references the fact that for any matrix $A$ and any non-zero $b$ we have $$\lim_{k\rightarrow \infty} A^k b = v$$ Where $v$ is the eigenvector of $A$ with the largest eigenvalue. The proof ...
0
votes
2answers
92 views
How do I write this matrix in Jordan-Normal Form
I have the matrix $A=\begin{pmatrix}2&2&1\\-1&0&1\\4&1&-1\end{pmatrix}$, I want to write it in Jordan-Normal Form. I have $x_1=3,x_2=x_3=-1$ and calculated eigenvectors ...
3
votes
1answer
84 views
Diagonalizable Matrix $A^2$
How can I find a matrix $A$ such that $A^2$ is diagonalizable but $A$ is not?
I have tried many different ways, but to no avail. Is there something that I am missing in the question that gives a ...
1
vote
1answer
46 views
Expressing a matrix as an expansion of its eigenvalues
This shouldn't be too difficult but I can't find a satisfactory proof.
Show that a real, symmetric matrix A satisfying the eigenvector
equation $Au_{i} = \lambda u_{i}$ cam be expressed as an ...
1
vote
1answer
101 views
How to calculate left and right eigenvector corresponding to the zero eigenvalue.
I'm working on $8\times8$ matrix resulting from the Jacobian of $8$ differential equation of a disease model evaluated at disease free equilibrium. I needed to get the left and right eigenvectors ...
4
votes
1answer
120 views
What is the difference between first and second right eigenvectors of a row stochastic matrix and their meaning?
In an $n\times n$ non negative row stochastic matrix (rows sum up to 1).
The entries of the stochastic matrix I have represent directed links between countries.
Why is the first right eigenvector a ...
4
votes
1answer
79 views
eigenvalues of the sum of a matrix with known eigenvalues and a diagonal matrix
Suppose $B = A+D$, where all the eigenvalues of $A$ are already known and $D$ is a diagonal matrix, how to compute the eigenvalues of B without diagonalizing $B$ directly?
1
vote
2answers
78 views
trace of the inverse of a matrix times another matrix
I do know generally $\text{trace}(A^{-1}B)\not= \sum_i \lambda_{B_i}/\lambda_{A_i}$,
where $\lambda_{A_i}$ and $\lambda_{B_i}$ are the corresponding eigenvalues of matrix $A$ and $B$ respectively,
...
3
votes
1answer
47 views
eigenvalues of block matrix with the eigenvalues of one block already known
Give a matrix which can be decomposed into 4 parts
$B = \left[\begin{matrix}A &I \\ -I &0\end{matrix}\right]$
where $I$ denotes the identity matrix and $0$ is a zero matrix.
It's easy to ...
4
votes
3answers
185 views
Eigenvalues and Eigenvectors of Large Matrix
Computing eigenvalues and eigenvectors of a $2\times2$ matrix is easy by solving the characteristic equation. However, things get complicated if the matrix is larger.
Let's assume I have this matrix ...
4
votes
1answer
145 views
Construct a matrix transform
consider
$\frac{dx}{dt} = Ax$ where $A$ is the matrix
$$
\begin{bmatrix}
1 & 0 & 1 \\
0 & 0 & -2 \\
0 & 1 & 0 \\
\end{bmatrix}
$$
...
1
vote
1answer
78 views
Question on generalized eigenspaces of commuting matrices
The following question came up as a though while I was reading. I cannot see how to proceed on it.
Let us have $M_1,\ldots,M_n$ be commuting matrices. I know that that the generalized eigenspaces are ...
1
vote
1answer
79 views
Can we always recover a matrix from its eigenvalues and eigenvectors?
If we're given all the eigenvalues of a square matrix $A$ and the corresponding eigenvectors of each eigenvalue, then in what case(s) is it possible theoretically to recover $A$ from this much ...
0
votes
0answers
107 views
What are the eigenvectors of this normal matrix?
Suppose $A$ is a real $n\times n$ normal matrix, and consider the matrix $$B=(I_n-\theta A)^{-1}(A+A^T)(I_n-\theta A^T)^{-1},$$ where $\theta$ is a real scalar such that $(I_n-\theta A)$ is ...
2
votes
1answer
29 views
Transforming a tensor representing matrix to the eigen system
Say I have some mapping in 2D t(v) = ... that is a tensor. I can find a matrix 2x2 T that represent this tensor, and find the ...
1
vote
2answers
54 views
Getting a feel for the transformation A on vector x which lies outside of any eigenspace
In one of his videos, after 13:25 Sal starts to talk about the interpretation of the eigenvectors and how they relate to a vector $x$ being transformed by the ...
3
votes
1answer
152 views
diagonalization of a bisymmetric matrix
Is there some way to easily diagonalize a rank $n$ bisymmetric Toeplitz matrix with only zeros on its main diagonal? Direct calculation is out of the question, I need some trick... thanks!
Addendum:
...
2
votes
2answers
58 views
Stable method to compute $A^n$ for this defective matrix $A$?
I'm looking for a stable method to compute $A^n$, where $A$ is the following defective $12 \times 12$ matrix:
$$A = \left(\begin{array}{cccc|cccc|cccc} \frac{1}{2} & \frac{1}{2} & 0 & 0 ...
1
vote
1answer
40 views
There are $n!$ different unitary similarities that give $L_i \sim L_j$. Or are there more?
Given a matrix $L$ of dimension $n \times n$ which is lower triangular, and with distinct elements along the diagonal, there is a way to unitarily transform it into a different lower triangular matrix ...
1
vote
1answer
75 views
Eigenvector of A to given Eigenvalue which requires row swapping to get reduced echelon form
Given the Matrix $$A = \left(\begin{matrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
-1 & -3 & -3
\end{matrix}\right)$$
calculate the eigenvalues and the corresponding eigenvectors of ...
1
vote
1answer
113 views
Verifying Orthogonality of Eigenvectors
How do you 'verify' the orthogonality of the eigenvectors of a matrix, let's say ${\bf A}$ , for example? I came across the result that a matrix ${\bf A}$ has orthogonal eigenvectors if ${\bf ...
0
votes
1answer
83 views
Problem with null Eigen vectors.
Please help me calculate the Eigen vectors of this matrix.
$$\begin{pmatrix}
3 & 0 & 1\\
1 & 3 & 0\\
0 & 1 & 3
\end{pmatrix}$$
The first vector comes out to be null, no clue ...
2
votes
2answers
578 views
Minimal polynomials and characteristic polynomials
I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices.
When are the minimal polynomial and characteristic polynomial the ...
2
votes
1answer
408 views
Upper Triangular Form of a Matrix
I am trying to find the upper triangular form of $B$ and an invertible matrix $C$ such that $B=C^{-1}AC$ where A is given by the following:
$A = \pmatrix{1&1\\
-1&3}$
The ...
1
vote
1answer
74 views
Find eigenvalues and eigenvectors: strange case
Why is this expression:
$$\begin{pmatrix}
\frac{k+mg}{l} & -k\\
-k & \frac{k+mg}{l}
\end{pmatrix} \begin{pmatrix}
\rho_1\\\rho_2
\end{pmatrix}=\omega^2\begin{pmatrix}
m & 0\\
0 & m
...
3
votes
3answers
204 views
What is the definition of a generalized eigenvector?
I am studying Generalized Eigenvectors. It seems that we can define them as $\mathbf{p}_i$ in this equation:
$$ (\mathbf{A}-\lambda\mathbf{I})^{k}\mathbf{p}_i = \mathbf{0} $$
in which $k$ is the ...
1
vote
2answers
112 views
A matrix without cube root
On the final exam of my linear algebra class, the instructor asked us to compute the cube root of matrix $$A = \begin{pmatrix}3 &2 & 0 \\ -4 &-3 &0\\ 0 &0& 8 \end{pmatrix}$$ I ...
