# Questions tagged [eigenvector]

The eigenvector tag has no usage guidance, but it has a tag wiki.

285
questions

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### Derivatives of a rotation tensor intended to rotate bases of a symmetric 2nd order tensor [closed]

Let $A = \sum_i \lambda_i N_i \otimes N_i$ and $B= \sum_j \mu_j n_j \otimes n_j$ , be the two symmetric and independent 2nd order tensors and their respective spectral decompositions. Let R be the ...

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51
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### Different versions of Courant–Fischer theorem

Let $A \in \mathcal{M}_n(\mathbb{R})$ be a symmetric matrix and $\lambda_1 \leq \lambda_2\ldots\leq\lambda_n$ be its real eigenvalues taken with multiplicities. Let $1\leq i_1 \leq i_2\ldots\leq i_k\...

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63
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### Smallest eigenvalue of certain PD matrix decreases under sparse perturbation

Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...

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2
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351
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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& \...

2
votes

1
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259
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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 ...

2
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0
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129
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### Convergence of eigenfunctions

In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...

2
votes

1
answer

86
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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 ...

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0
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48
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### Random matrices: Relation between leading eigenvector and a vector in culumn space

Let $X$ be a $n\times n$ symmetric matrix with iid zero-mean random entries on and above the diagonal. Denote by $v$ the eigenvector corresponding to the largest eigenvalue of $X$. Let $a$ be a fixed $...

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176
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### Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart

I am trying to understand the connection between the eigenspace of the continuous operator
$$
H(x,y) = \frac{1}{x+y}
$$
which is nothing but the square of the Laplace operator, and its discrete ...

3
votes

1
answer

143
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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 \...

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### Nonnegative eigenvector form the point of view of variational characterization

Let $\Omega$ be a smooth domain in $\mathbb{R}^N$ $(N\geq 3)$. We denote by $G(x, y)$ the Green function for the boundary value problem
$$
-\Delta_x G(x, y)=c_n \delta(x-y) \quad \text { in } \Omega, ...

10
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0
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597
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### Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$

My question seems elementary, yet I could not find the solution after working on and searching for several days...
I'd like to find the eigenfunctions of a simple integral kernel:
\begin{equation}
\...

2
votes

1
answer

215
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### Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$

Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set
$$\tag{1}
\int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...

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105
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### Formula for the kernel of an operator

Let $\mathcal H$ be a Hilbert space and let $O$ be an operator. Obviously $M=O^\dagger O$ is a semi-positive definite operator and $v\in\ker M$ if and only if $v\in\ker O$. Therefore it seems to me ...

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2
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312
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### Eigenvectors of a non-symmetric rank-one update of a symmetric matrix

I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+uv^\top$ where $u$ $(n\times 1)$ and $v$ $(n\times 1)$ are column vector. Also, $A=yy^\top$ with $...

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0
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318
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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 ...

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0
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152
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### Eigenvalues/eigenfunctions of a diffusion generator

Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by:
$$L\phi := \lambda_1 \partial_{R_1}(R_1 \...

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66
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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 $X=(x_{ij})\in M_n(\mathbb{R})$ ...

3
votes

1
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59
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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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146
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### Construct a permutation matrix from some eigenvectors and eigenvalues

Given $n$ orthonormal vectors $v_1, \dots, v_n \in \mathbb R^d$, where $d > n$, we can show that there are many orthogonal matrices $X$ of size $d$ such that $v_1, \dots, v_n$ are eigenvectors of $...

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1
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144
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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,...

2
votes

1
answer

108
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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 $Te_k=e_{k-1}$ ...

2
votes

1
answer

179
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### Are these the only first eigenfunctions on a hemisphere?

Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$...

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0
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140
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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

125
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### 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{...

2
votes

1
answer

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### Eigenvalues of a rank-one update of a symmetric matrix

I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+xx^\top$ where $x$ $(n\times 1)$ is a column vector.
and also $A=yy^\top$ with $y$ a $(n-1)$ rank ...

1
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0
answers

180
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### Eigenvalue decomposition of normalized adjacency matrix

Let $A$ be an adjacency matrix of undirected graph $G$, where $G$ is a connected graph. The normalized adjacency matrix is defined as $\hat{A}=D^{-1/2}AD^{-1/2}$, where $D$ is degree matrix of graph $...

1
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0
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72
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### Are eigenfunctions of the Dirichlet problem for the Laplace equation uniformly bounded?

Let $Q\subset \mathbb R^n$ be a bounded domain with boundary $\partial Q\in C^\infty$ and $\varphi_1,\varphi_2,\ldots$ are eigenfunctions of the Dirichlet problem for the Laplace equation in $Q$ ...

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1
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195
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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 ...

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0
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99
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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 ...

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0
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95
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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 $T$?

1
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0
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58
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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]^\...

1
vote

1
answer

77
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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.
...

3
votes

1
answer

261
views

### Are Neumann Laplacian eigenfunctions in $C(\overline{\Omega})$?

Consider that $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ (in the distributional sense) such that for some $\lambda>0$ we have that:
$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\...

2
votes

1
answer

70
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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
$$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...

1
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1
answer

94
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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 & \...

0
votes

1
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74
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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.
$$...

3
votes

0
answers

111
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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 $...

3
votes

1
answer

181
views

### 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?

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129
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### What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^{\frac{n-1}{2}}$ when $n$ is odd?

Suppose that $n$ is odd. The eigen values/eigenvectors of the skew-circulant matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ are successfully computed in this post.
Q. What are ...

2
votes

1
answer

298
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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?

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vote

1
answer

137
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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?

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vote

2
answers

316
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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 & ...

6
votes

0
answers

170
views

### Measurability of eigenvalues-eigenvectors of a positive compact operator

Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$.
By the spectral theorem, given $a \in A$, there are scalars $...

0
votes

0
answers

240
views

### 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}=...

3
votes

0
answers

47
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### What is the supremum of 1-dim Hausdorff measure of the nodal set of Neumann eigenfunction $u$ for planar convex domain

All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote $\Sigma=\{u(x)=0\}$ be the nodal set of a ...

3
votes

2
answers

366
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 & &\\
& &...

0
votes

0
answers

127
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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 \...

0
votes

1
answer

291
views

### Are there zero entries in the eigenvector corresponding to a simple eigenvalue?

For a real symmetric matrix $M$ and a simple eigenvalue $\lambda$, under which conditions the corresponding eigenvector has no zero entries? Perhaps, this is unconditional and one can provide a proof?

4
votes

1
answer

166
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### Least squares problem with left and right unknowns

For $i=1,...,n$, let $b_i$ be a scalar and $A_i$ be an $k\times l$ matrix. Is there a closed form solution for the following problem assuming $n>k+l$?
$$\min_{x\in \mathbb{R}^k ,y\in \mathbb{R}^l} \...