All Questions
Tagged with linear-algebra eigenvector
157 questions
0
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157
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Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?
I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components".
I noticed that in the accepted ...
- 23
1
vote
0
answers
80
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Inequality involving random vectors and absolute values
Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
3
votes
1
answer
99
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Eigenvectors of $P^\top P$ for 0/1 matrices $P$
Let $P$ be an $m \times N$ matrix of zeros and ones (think $N \gg m$), and let $\mathbf{u} \in \mathbb{R}^N$ be a unit vector satisfying $P^\top P\mathbf{u} = \lambda^2 \mathbf{u}$ for some $\lambda &...
- 161
0
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0
answers
101
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Eigenvectors of tridiagonal hermitian matrix
In my paper, I investigate the coordinates of the eigenvectors of a hollow tridiagonal hermitian matrix, which is defined as:
\begin{align*}
Q_n =
\begin{pmatrix}
0 & q_{1,2} & 0 & 0 & ...
- 1
0
votes
0
answers
77
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Eigenvalues of N×N correlation matrices as N tends to infinity
I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero.
...
4
votes
1
answer
314
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Some but not all eigenvectors mutually orthogonal
Suppose an $n\times n$ matrix has real entries and has $n$ real eigenvalues and its eigenvectors span $\mathbb R^n.$ Are there any interesting conditions under which $k$ of its eigenvectors are ...
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2
answers
397
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How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
Consider the following $(2d+1)\times (2d+1)$ matrix:
$$
A = \begin{pmatrix}
0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\
\frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...
- 25
2
votes
1
answer
278
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Continuity of eigenvector of zero eigenvalue
Wonder whether anyone has an idea on showing the following or to point out that it is not true:
Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
- 69
2
votes
1
answer
142
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Lipschitz continuity of eigenprojections
This question has the same flavor of this and this questions, but asks for something stronger.
Assume that
$A$ is a symmetric $n \times n$ matrix,
$H$ is a $n \times n$ perturbation matrix.
Moreover ...
- 21
3
votes
1
answer
155
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Does this matrix equation always have a solution?
Let $\{A_i, i\ge 3\}$ be the matrices whose columns represent numbers from $0$ to $2^i-1$ in binary form. For example,
$A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \...
- 235
1
vote
0
answers
329
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The geometrical multiplicity of the nilpotent matrices
The following point is well-known in the literature.
Theorem. Let $A$ be a non-negative matrix in $M_n(\mathbb{R})$. If $A$ is nil-potent, there is a permutation matrix $P$ such that $P^tAP$ is ...
- 4,058
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0
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74
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Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix
Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
- 4,058
3
votes
1
answer
67
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Why does the normalization term disappear when computing the MLE of decomposed Gaussians
Computing the Maximum Likelihood Estimator of Gaussians in arbitrary finite Hilbert spaces seems no easy task and I must admit to lamentably fail at it. The classical theory most often relies on ...
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1
vote
1
answer
198
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Directed graph whose adjacency matrix admits only 0 as eigenvalue
Let $G$ be a directed graph and let $P_i$
be its vertices. Let $A$
be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$
if and only if there is a directed edge from $P_i$
to $P_j$, ($a_{i,...
- 4,058
2
votes
1
answer
112
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The eigenvectors of adding a particular rank one matrix to the circulant matrix
Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.
Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
- 4,058
1
vote
0
answers
179
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QR algorithm for eigenvalues and eigenvectors of large symmetric matrices
I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices,
My initial thought was to use Householder transformation with a Wilkinson shift ...
1
vote
1
answer
136
views
Matrix transformation that always works?
Consider the matrix
$$A_2:= \begin{pmatrix} a & b_1 \\ b_2 & a\end{pmatrix}.$$
Let $\sigma_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$, then
$$\sigma_2 A_2 \sigma_2 = \begin{...
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votes
1
answer
225
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Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
- 103
0
votes
0
answers
107
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The eigenstructure of the symmetric tridiagonal matrix whose entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and $$a_{1,2}=\cdots=a_{n-1,n}=1$$
Suppose that $A=(a_{kl})_{k,l=1}^n$ is a symmetric tri-diagonal matix in $M_n(\mathbb{R})$ whose diagonal entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and
$$a_{1,2}=\cdots=a_{n-1,n}=1$$
Any approach to ...
- 4,058
0
votes
0
answers
124
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Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same
Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with
$$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$
How can we compute the eigenvectors of $T$?
- 4,058
1
vote
0
answers
66
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CLT of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional random matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\...
- 284
1
vote
1
answer
84
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Asymptotic property of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top.
...
- 284
2
votes
1
answer
76
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The eigenvalues of the product $WD$ for some particular matrices
Let $D$ be a diagonal matrix in $M_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0
Suppose that $e_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is
$$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...
- 4,058
1
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1
answer
107
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Convergent condition of the high-dimensional submatrix of some orthogonal matrix
Let $\mathbf{V}$ be a $p\times p$ orthogonal matrix (i.e., $\mathbf{V}\mathbf{V}^\top = \mathbf{V}^\top \mathbf{V} = \mathbf{I}$) whose columns are
$$
\mathbf{V} = \begin{bmatrix} \mathbf{v}_1 & \...
- 284
0
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1
answer
75
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The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$
Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$...
- 4,058
3
votes
0
answers
115
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Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
- 4,058
3
votes
1
answer
181
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The proof of the invertibility of $\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$
Suppose that $n$ is even. Any suggestion/appraoch to prove that $S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$ is invertible?
- 4,058
3
votes
1
answer
370
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The eigenvalues of the matrix $\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$
What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ when $n$ is odd?
- 4,058
1
vote
1
answer
145
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The dimension of the eigenvector space of non-negative irreducible matrices
Let $A$ be an irreducible non-negative matrix. Is it true that the eigenvectors of $A$ can span the $R^n$ ?
Or are all the eigenvalues of $A$ distinct?
- 115
1
vote
2
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446
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Transforming matrix to off-diagonal form
I wonder if one can write the following matrix in the form $A = \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix}.$
The matrix I have is of the form
$$ C = \begin{pmatrix} 0 & a & b & ...
- 536
0
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0
answers
273
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Finding the eigenvectors of a submatrix
Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by,
$b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$.
$b_{n+k,l}=...
- 4,058
3
votes
2
answers
394
views
Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix
The question is the following: given a matrix
$$A=\begin{pmatrix}
1& 2 & & & & \\
1& 0& 1 & & & \\
& 1& 0& 1 & &\\
& &...
- 145
0
votes
0
answers
149
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Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
- 1
10
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2
answers
473
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Support of eigenvectors
Consider the $N$ by $N$ matrix
$$M_N= \begin{pmatrix} 1+3\lambda & -1-2\lambda & - \lambda & 0 & 0 &0 &0\\
-1-2\lambda & 2+3\lambda & -1 & -\lambda & 0 & 0 &...
- 192
2
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1
answer
141
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On the eigen vectors of a diagonalizable matrix
Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$.
Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...
- 4,058
1
vote
0
answers
274
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Find the eigenvectors from the QR algorithm in the unsymmetric case
It is possible to find many references describing the QR Algorithm with more or less refinements to approximate the eigenvalues of a square matrix $A\in\mathbb{R}^{n\times n}$.
I implemented a version ...
- 11
2
votes
1
answer
1k
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Diagonalizing a symmetric block matrix
Let us consider the matrix
$$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$
then this matrix has eigenvalues $a\pm \sqrt{c^2+b^2}.$
Now, let us consider a block matrix
$$ A = \begin{...
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5
votes
0
answers
208
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Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
- 73
4
votes
2
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372
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How close to uniform are Perron-Frobenius eigenvectors?
Let $A=(a_{i,j})$ be a square matrix with non-negative entries. (Assume $A$ is symmetric, if it helps.) Let $v$ be a Perron-Frobenius eigenvector. What do we need to assume about $A$ in order to have ...
- 20.2k
1
vote
0
answers
474
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When must an eigenvector have only non-negative entries?
What would be a reasonable sufficient condition on a real symmetric matrix that would force its eigenvector with largest eigenvalue (or one of its eigenvectors with maximal eigenvalue) to have only ...
- 20.2k
1
vote
1
answer
176
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Derivative of eigenpair with respect to matrix
Suppose that $A$ is real and symmetric matrix (or tensor) of dimension $3 \times 3$, with its spectral decomposition
$$A = \sum_{i=1}^3 \lambda_i\ n_i\otimes n_i$$
where $\lambda_i$, $n_i$ and $\...
- 13
0
votes
0
answers
88
views
Fast decay of eigenvector elements
Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest ...
- 47
10
votes
2
answers
614
views
Lower eigenvectors of nonnegative matrices with zero trace
Let $A$ be an $N\times N$ nonnegative matrix with all diagonal entries equal to zero and such that there is $n_0$ such that all entries of $A^{n_0}$ are strictly positive. Let $\lambda_1,\ldots, \...
- 1,897
2
votes
1
answer
1k
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Derivative of eigenvectors of an Hermitian matrix
In the question "Derivative of eigenvectors of a matrix with respect to its components", Liviu Nicolaescu has provided an answer valid for a real matrix. As outlined in the following, the ...
- 23
2
votes
1
answer
294
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Find a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations
I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows represent the same parameters in the 2 matrixes).
Now I would like to make the cross-correlations ...
4
votes
1
answer
747
views
Condition number for matrix of eigenvectors of a diagonalizable matrix
Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$.
For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} ...
- 41
1
vote
1
answer
143
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Requirements for finite backward derivatives of degenerate eigenvectors
A matrix, $\mathbf{A}(\theta)\in\mathbb{R}^{n\times n}$, has elements that depend on a parameter $\theta$. The $j$-th eigenvalues and eigenvectors of the matrix are denoted as $\lambda_j$ and $\mathbf{...
- 111
3
votes
1
answer
1k
views
Calculating second derivatives of eigenvectors of a matrix with some degenerate eigenvalues
Given real symmetric matrix $\mathbf{M}$ with eigenvalues $\lambda_i$ and eigenvectors $\mathbf{v}_i$, the derivative of an eigenvector is $$\dot{\mathbf{v}}_i = \sum_{j \ne i} \frac{\mathbf{v}_j \...
- 33
5
votes
1
answer
1k
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Real eigenvectors of complex matrices
Let $A$ be a nonsingular complex $(3 \times 3)$-matrix (that is, an element of $\mathrm{GL}_3(\mathbb{C})$).
Then what are some of the best-known criteria which guarantee $A$ to have real eigenvectors ...
- 4,547