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11 votes
2 answers
2k views

Inverse of a small submatrix

Let $A$ be a large matrix (say, $1000 \times 1000$), and let $\mathcal I = \{2,3,5\}$ be a set of row/column indices. Let $(A^{-1})_{\cal I \times I}$ denote the submatrix of $A^{-1}$ that consists of ...
John Smith's user avatar
11 votes
0 answers
313 views

Jaffard's theorem - finite matrices

For infinite matrices, Jaffard's theorem states that if $(A(k,l))_{k,l\in \mathbb{Z}}$ is invertible and satisfies $$ A(k,l) \leq C (1+\left|k-l\right|)^{-r}, $$ for some $C>0$, then $$ A^{-1}(k,...
Ozzy's user avatar
  • 393
9 votes
2 answers
894 views

Inverse of special upper triangular matrix

Consider the following $n \times n$ upper triangular matrix with a particularly nice structure: \begin{equation}\mathbf{P} = \begin{pmatrix} 1 & \beta & \alpha+\beta & \dots & (n-3)\...
dff's user avatar
  • 230
4 votes
2 answers
1k views

Derivative of pseudoinverse with respect to original matrix

I have been trying to find an analytical expression for the following: $\frac{\partial {X^{+}}}{\partial {X}}$ In my case, $X$ has a constant rank. I've found the formula for differentiating a ...
Tarrare's user avatar
  • 143
3 votes
1 answer
349 views

Inverting (via Taylor expansion) a sum of (rank-deficient) skew-symmetric matrix and (rank-deficient) Diagonal matrix

I have the following problem: A matrix $C\in \mathbb{R}^{2N}$, where $C=\epsilon A+D$ $\epsilon A=(C-C')/2$ is skew symmetric with "block" anti-diagonal structure of size 4. $ D=(C+C')/2$ (Diagonal ...
mystupid_acct's user avatar
3 votes
0 answers
73 views

Regularity of Moore-Penrose pseudo-inverse

Let $k\in\mathbb{N}\cup\{0\}$, let $\Omega\subseteq\mathbb{R}^n$ be open, connected and let $G\in C^k(\Omega;\mathbb{R}^{n\times n})$ satisfy $$ \operatorname*{rank}G(x)= \operatorname*{rank}G(y),\...
Tatin's user avatar
  • 895
1 vote
1 answer
896 views

sign-flipping inverse

Consider this matrix: $Z=\begin{bmatrix}23.9 & -7 & -17 \\\\ -7 & 23.9 & -17 \\\\ -17 & -17 & 33.9 \end{bmatrix}$ Its inverse is entrywise negative (you can check...) and ...
Felix Goldberg's user avatar
1 vote
1 answer
151 views

How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$

Let $A\in \mathbb{R}^{m\times n}$, $m>n$, $rank(A)=n$, and $\forall 1 \leq i \leq m, 1 \leq j \leq n, A(i, j)>0$, $y=[1, 1, 1, ..., 1]^T$. Let $\beta=A(A^TA)^{-1}A^Ty$, how to prove that each ...
Songqiao Hu's user avatar
1 vote
2 answers
2k views

Bound on smallest entry of inverse matrix

For a symmetric, invertible matrix $A=(a_{ij})\in \mathbb{R}^{n\times n}$ with (at least two) nonzero off-diagonal elements, is it possible to bound in absolute value the smallest entry of its inverse ...
user47575's user avatar
1 vote
1 answer
118 views

A sine type Chebyshev system

A sequence of real functions $\{\phi_1,\cdots,\phi_n\}$ is called a Chebyshev system on an interval $I\subseteq\mathbb{R}$, if any real linear combination $\sum_{l=1}^n a_l\phi_l$ has at most $n-1$ ...
ABB's user avatar
  • 4,058
0 votes
1 answer
127 views

Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix,$A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$

This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$. As known, ...
Songqiao Hu's user avatar
0 votes
1 answer
91 views

Matrix quantization and effect on singular values

Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$, and let $\sigma_1(X),\dots \sigma_n(X)$ denote the singular values of $X\in \{A,B\}$. Do we have upper and lower bounds for $$ \| \sigma_i(A)-\...
ABIM's user avatar
  • 5,405
0 votes
0 answers
957 views

Diagonal of the inverse of a 6x6 symmetric partitioned matrix

Let $$M = \begin{bmatrix} A & B \\ B & C \end{bmatrix}$$ in which $A$, $B$ and $C$ are $3 \times 3$ matrices being also symmetric. In fact, they are quite similar, just differing on a single ...
Martin's user avatar
  • 1