All Questions
12 questions
2
votes
1
answer
112
views
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
0
votes
0
answers
108
views
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
125
views
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
3
votes
0
answers
115
views
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
0
votes
1
answer
75
views
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
2
votes
1
answer
141
views
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
2
votes
0
answers
301
views
Eigenvector of Hadamard matrix functions
Let $X\in\mathbb{R}^{n\times n}$ with SVD $X=UDV^T$. Are there known results regarding the eigenvectors of $Y=X^{\odot g}$? I am mainly interested in simple functions such as $g(z)=z^2$, i.e. $Y_{ij}=...
- 21
7
votes
1
answer
6k
views
Eigenvectors as continuous functions of matrix - diagonal perturbations
The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
- 2,222
1
vote
1
answer
437
views
Eigenvector of a nonnegative matrix in closed form
Consider $n\times 1$ vector $\alpha = (\alpha_{1}, ..., \alpha_{n})$, where $0<\alpha_{i}<1$, and $\sum_{i=1}^{n}\alpha_i = 1$. Construct the $n\times n$ zero-diagonal matrix $A$ with $(i,j)$-th ...
- 1,538
0
votes
2
answers
737
views
Eigenvalues of an amplification matrix
Let $A$ and $B$ square real matrices.
I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1.
Can we say something about the eigenvalues ...
- 1
3
votes
1
answer
264
views
When is this matrix singular?
Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.
1) For $t_k=k$, what is the condition on $\...
- 2,251
1
vote
1
answer
136
views
Any generic way to move a psd matrix to its neighbors?
Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...
- 363