Questions tagged [eigenvalues-eigenvectors]
Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.
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Eigenvalues of $8\times8$ non-diagonal matrix $A$ satisfying $A^2 - 6A + 9I_8 = 0_8$
In an exam today I had the following problem:
Let $A \in M_{8}(\mathbb{R})$ ($8\times8$ matrix with real entries) satisfying $$A^2 - 6A + 9I_8 = 0_8$$ Find the eigenvalues of $A$.
My approach
If $\...
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Diagonalization of complex, symmetric but defect matrices
Real symmetric matrices (not Hermetian $M^\dagger = M$, but symmetric $M^T = M$) can according to Autonne–Takagi factorization, as described here, be diagonalized. Thus, we can find a unitary matrix $...
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Can the block-Lanczos algorithm possibly converge faster than the single-vector Lanczos?
We use the Lanczos algorithm for finding eigenvalues and eigenvectors of large sparse real matrices to model atomic nuclei. However, for heavier nuclei and their higher energy states, the matrix ...
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Where are the formulas for frequency $\omega=\sqrt{\lambda_1 \lambda_2}$, period $T=2\pi/(\sqrt{\lambda_1 \lambda_2)}$
Where are the formulas for frequency $\omega=\sqrt{\lambda_1 \lambda_2}$, period $T=2\pi/(\sqrt{\lambda_1 \lambda_2)}$ got from?
For e.g. Lotka-Volterra system and associated Jacobian and eigenvalues/-...
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can the eigenvalues of a symmetric tensor be complex?
Let $T$ be a fully symmetric tensor of rank $3$ and size $N$.
Using the following definition of eigenvalues, let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that:
\begin{equation}
\sum_{jk}^...
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best way to show tr(AB) = tr(BA) for non square A and b matrices
I have matrices $A \in \mathbb{K}^{n \times m}$ and $B \in \mathbb{K}^{m \times n}$
What is the best way to prove that tr(AB) = tr(BA). I found a prove in Matrix Analysis by Horn and Johnson, but they ...
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Can I use Weinstein–Aronszajn identity to show that the eigenvalues of AB and BA are the same?
I have the matrices $A \in \mathbb{K}^{n \times m}$ and $B\in \mathbb{K}^{m \times n}$ .
Can I use the Weinstein–Aronszajn identity to show that $AB$ and $BA$ have the same non-zero eigenvalues? If so,...
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Eigenvalues of a linear transformation $M_{2,2}\ \rightarrow M_{2,2}$
I took linear algebra years ago, and for various reasons I'm going back to re-learn it. In that process, I came across the following question in an old exam from the course I once took:
A linear ...
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Finding the characteristic polynomial of this matrix
I have a question about this post.
How can we know that the geometric multiplicity of the eigenvalue $\lambda=0$ is $n-1$?
I get that $0$ is an eigenvalue of $A$ because $\textrm{det}A=0$, but I can't ...
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Calculating eigenvectors of a 3x3 symmetric matrix from Cayley–Hamilton theorem?
Let's say I have a 3x3 real symmetric matrix M and I have its three eigenvalues $\alpha_1, \alpha_2, \alpha_3$ and that $\alpha_1 < \alpha_2 < \alpha_3$. (Notice all eigenvalues are different).
...
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conditions for which matrix has all eigenvalues inside unit circle or on the unit circle
$\mathbf{M}=\begin{bmatrix}
\mathbf{y}&0&0&\cdots&0&0&\mathbf{x}\\
\mathbf{I}&0&0&\cdots&0&0&0\\
0&\mathbf{I}&0&0&0&0&0\\
0&...
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Two different decomposition of scaled identity matrix
Let $\mathbf{\Sigma} = (1/2)I_2$ where $I_2$ is the $2 \times 2$ identity matrix. Give two different selections for $\lambda_1, \lambda_2 \in \mathbb{R}_{\geq 0}$ with $\lambda_1+ \lambda_2 =1 $ and $\...
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Eigenvalues of a particular block circulant matrix
I need to compute all the eigenvalues of the following block-circulant matrix for a research. Can anyone help me compute the eigenvalues of the following matrix?
$$\left[\begin{array}{l}2I&-I&...
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Does eigenvalues of matrix change after multiplication by unitary matrix?
For a matrix $A \in \mathbb{C}^{n \times n}$, does multiplication by a unitary matrix $U$ change the eigenvalues of $A$? So for:
$$Ax = \lambda x \qquad \mathrm{and} \qquad AUy = \mu y $$
does $\...
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Eigenspace of a specific eigenvalue
I am computing eigenvalues and eigenfunctions of Laplacian on a unit square $[0,1]^{2}$ numerically.
Consider the eigenvalue problem with the Dirichlet boundary condition that is, $$L u(x, y) = \...
user886636
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How does permuting rows and columns change the eigenvectors of a matrix?
Let $A\in\mathbb{R}^{n\times n}$ be any matrix and let $P\in\mathbb{R}^{n\times n}$ be a permutation matrix.
Left multiplication of $A$ by $P$ corresponds to permuting the rows of $P$. On the other ...
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Finding eigenvalue of a matrix expression
Consider the following matrix
$A=\left[\begin{array}{lll}1 & 0 & 0 \\ 2 & 3 & 0 \\ -3 & 1 & -2\end{array}\right]$
How can I find the eigenvalues of $3 \mathrm{~A}^{3}+5 \mathrm{...
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The Eigenvalues of a block matrix with nonzero blocks
Let $M$ be a block matrix:
$$
M = \left(\begin{array}{cc}
A & B \\ C & D
\end{array}\right)
$$
Is there any relation between eigenvalues of $M$ and eigenvalues of matrices $A, B, C,$ and $D$?
...
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Proving that $1+λ^2$ is an eigenvalue of $I+A^2$ where $λ$ is an eigenvalue of matrix $A$
So $Ax = λx$;
$A(Ax) = λ(Ax) \to (A^2)x = (λ^2)x$
I kind of dont know how to get the $1$ and $I$ here...
Any help is appreciated
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Given some of the eigenvalues and corresponding eigenvectors reconstruct the entire matrix.
Given a symmetric matrix $A$ that has eigenvalues $4, 3, 2,$ and $2$ and the eigenvectors belonging to the eigenvalues $4$ and $3$. Provide a procedure
to reconstruct the entire matrix.
Since the ...
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Finding an eigenvector given other eigenvectors for JNF
I want to find the Jordan Normal Form of the matrix
$$M=\begin{pmatrix}
19 & -7 & -1 & 5 & -7 \\
3 & 9 & -1 & 3 & -5 \\
-...
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number of matrices with a single nullspace point
Suppose I fix a vector $v\in \mathbb{R}^n$. I want to count the number of maximal rank (in this case $n-1$) linearly independent matrices $A_1,\ldots,A_m$ for which $v$ lies in the nullspace of $all$ ...
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How to find the eigen vector of a system that does not have similar equations?
Suppose I am trying to solve for an eigenvector provided
my matrix
$$A=\left(
\begin{array}{cc}
2 & 7 \\
7 & 2 \\
\end{array}
\right)$$
eigenvalues are $9$ and $-5$
How can I get an ...
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Eigenvector and eigenvalue for power of matrix, will eigenvectors remain the same?
I understand that if you raise a matrix $ A $ to let's say a power of 5, assuming that $ A$ is a 3 x 3 matrix and has 3 distinct eigenvalues, each of the eigenvalues will also be raised to the power ...
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Can two eigenvectors share one eigenvalue?
I apologize in advance for the mess. This is my attempt.
The matrix in question is
\begin{align} \textbf{A} = \frac{1}{100}\cdot\begin{pmatrix}
92 & 0 & -144 \\
0 & 100 & 0 \\
-144 &...
user883800
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Matrix Powers A^k - Eigenvalues and Eigenvectors - Chapter 6 Gilbert Strang
I confused about the formula below:
It's different from his lecture note. Why it's different? It doesn't matter the order?
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Can there be a different Eigen vector for a particular Eigen value?
Please see the photo.
Here, my answer came $k$
$\begin{bmatrix}
1\\
1\\
-1
\end{bmatrix}$
But their answer is given :
$k$
$\begin{bmatrix}
...
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PCA for image analysis - Eigendecomposition VS SVD
I have learned about two different ways of doing PCA, one using Eigendecomposition and the other (IMO more intuitive) using SVD. Although both are "just" a change of basis, I struggle to see ...
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Which of the statements is not necessarily true?
Let $A$ be a $3\times3$ matrix and $u, v, w$ be linearly independent vectors in $\mathbb{R}^3$ such that:
$Au = 2u, Av = 2v, Aw = 0$.
Which of the statements are NOT necessarily true?
Option 1: $w$ is ...
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Eigenvalues/Eigenvectors of operators on a Fock space
Suppose we have an operator defined by
$$S_0 = 1, \ S_n = 1_2^{\otimes n-1}\otimes X^n,$$
where $X = \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}$, $1_2$ is the $2\times 2$ identity matrix and $n&...
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Solving periodic ODE with finite difference having strange results...
I was wondering if you could help me understand.
I am trying to solve this eigenvalue problem numerically via finite difference (I've solved it already using a spectral method).
\begin{equation}
i \...
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Prove the existence of a sequence of eigenvectors
Let $(A_n),(B_n)$ be two sequences of $n\times n$ (growing sizes) symmetric real matrices. Suppose that their difference $\Delta_n:=A_n-B_n$ statisfies
$$||\Delta_n||\to 0 \text{ as } n\to \infty,$$
...
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Orthogonality with respect to random eigenvector
Let $A$ be an $n\times n$ real positive semi-definite random matrix. Let $x$ be the $n\times 1$ normalized eigenvector corresonding to its largest eigenvalue. Suppose I have a real random variable $z$ ...
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Finding eigenvectors of matrix of matrices
Let $A$ be a matrix with $\lambda_1,...,\lambda_n$ eigenvalues, and $v_1,...,v_n$ the corresponding eigvenvectors. Let $B= \begin{pmatrix} 0& A \\ A& 0 \end{pmatrix}$. It's known that the ...
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How does elementary row-multiplication change the left eigenvector of an irreducible nonnegative matrix?
Let $M$ be an irreducible nonnegative matrix and $E$ be some diagonal matrix where all diagonal elements are positive. It is clear that $EM$ is another irreducible nonnegative matrix. Letting $\rho(M)$...
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Better bounds on Perron Root
I have a (column) substochastic regular nonnegative matrix ($P$) obeying the Perron-Frobenius theorem. I am interested in the bounds on the dominant eigenvalue (Perron root). I know that I can bound ...
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maximum absolute column sum (column norm) of a matrix and its largest eigenvalue [closed]
Given a matrix $A\in\mathbb{R}^{n\times n}$. How to prove the following statement?
$$||A||_{L_1}\leq \sqrt{n}\sigma_{\max}(A),$$
where $||A||_{L_1} = \max_{i\in\{1,...,m\}}||\mbox{col}_i(A)||_1$ and $\...
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How can we evaluate the characteristic polynomial with a matrix as the parameter?
For any polynomial p(x) = $a_0+a_1x+· · ·+a_kx^k$
and any square matrix A, p(A) is defined
as p(A) = $a_0I + a_1A + · · · + a_kA^k$
. Show that if v is any eigenvector of A and $χ_A(x)$ is
the ...
user854990
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Eigenvalues of a matrix in relation to another matrix
Let $A,B \in \mathcal{M}_n(\mathbb{K}) $ (where $\mathbb{K} \in \{\mathbb{R},\mathbb{C} \} $) invertible matrices. Prove that if $\lambda = 0 $ is an eigenvalue of matrix $C$, where $C = BA - AB ~$ ...
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A semisimple Lie algebra has finitely many representation of a given finite dimension.
Let $\mathfrak{g}$ be a finite dimensional, semisimple Lie algebra, with Cartan subalgebra $\mathfrak{h}$. I want to show that $\mathfrak{g}$ has finitely many representations of dimension $n$, up to ...
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Upper and Lower Eigenvalue Bounds for $X^{T} A X$
Let $A \in \mathbb{R}_{+}^{n \times n}$ be diagonal with $i$th diagonal element $a_{ii}$ (i.e. all $a_{ii} > 0$), and let $X \in \mathbb{R}^{n \times p}$ be a matrix with rank $p$ where $n \ge p$ (...
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Expectation value of an operator
I am new to quantum mechanics and have some questions regarding expectation values of operators. I have the non-degenerate eigenstates $|\psi_n \rangle $ of an operator $A$. I have the normalized ...
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Quadratic Form of $A^2, A^3$
Assume $A\in \mathbb{S}^n$ is a positive semi-definite matrix, and denote its eigenvalues by $\lambda_1 > \lambda_2 \ge ... \ge \lambda_n \ge 0$. Let $w\in \mathbb{R}^n$ be a unit vector such that
$...
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Construction of characteristics polynomial
Suppose we have a matrix $A$ of order $n$. Let $\lambda$ be an eigenvalue of $A$ and $y$ be a corresponding eigenvector. Then from the relation $Ay = \lambda y$ we get some relation about the ...
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Find $\lim_{k\rightarrow \infty} A^k x$
Let $\displaystyle A=\left(\begin{matrix}.5&1\\0&.75\end{matrix}\right),x=\left(\begin{matrix}-2\\1\end{matrix}\right)$.
Find $\displaystyle \lim_{k\rightarrow \infty} A^k x$
Find $\...
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Finding eigenvalues of block type matrix
If I know the eigenvalues of matrix $A$, is there any way to calculate eigenvalues of $B$ using some tricks? Here $A$ is an $m\times m$, $O$ is the zero matrix and $I$ denotes $m \times m$ identity ...
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Explanation on Google’s PageRank is Webpages as Eigenvectors
Help understand what is the matrix A and the vector x discussed below.
Mathematics for Machine Learning Example 4.9
Google uses the eigenvector corresponding to the maximal eigenvalue of
a matrix A ...
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2
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Orthogonal on an ellipse
Consider two points $a \in \mathbb{R}^2$ and $b \in \mathbb{R}^2$ on the boundary of an ellipse such that $a^\top \Sigma^{-1} a = b^\top \Sigma^{-1} b$. I am trying to understand which points on the ...
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Degree of minimal polynomial when number of distinct eigenvalues are given
If a square matrix of order $ 10 $ has exactly $ 4 $ distinct eigenvalues, then degree of it's minimal polynomial is
At least $ 4 $
At most $ 4 $
At least $ 6 $
At most $ 6 $
$ 4 $ distinct ...
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Question about spectral radius of a positive matrix
I am learning the Perron-Frobenius theorem from some lecture notes. Let $X \in \mathbb{R}^{n}_{++}$ be a square matrix with each element being strictly positive. The theorem says that the spectral ...
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