# Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

92
questions

**1**

vote

**0**answers

38 views

### Fast computation of linear equation with row and column removed

Given $A, x, b$ and a linear system $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is full rank. Denote $A_{\backslash i}$ as a $(n-1)\times(n-1)$ matrix where $A$ removes its $i^{th}$ column and row, ...

**1**

vote

**1**answer

110 views

### Inverse of a larger matrix where the inverse of the submatrix is known

Let $A, A^{-1} \in \mathbb{R}^{n \times n}$ be known matrices. Suppose we have an invertible matrix $B \in \mathbb{R}^{(n+1) \times (n+1)}$ of the following form:
$$B = \begin{bmatrix}
A & ...

**0**

votes

**0**answers

37 views

### pseudo-inverse of short fat matrix

Consider the matrix $A \in \mathbb{R}^{m \times n}$ with $m < n$.
Is its pseudo-inverse $A^\dagger = (A^\top A)^{-1} A^\top$ computable ?
I'd expect not, because $A^\top A$ is not of full rank ( $...

**1**

vote

**1**answer

77 views

### Low-complexity method for sub-matrix inversion

Assume that $\mathbf{A}$ be an $N\times N$ matrix. We know that the complexity of the computation of matrix inversion is $\mathcal{O}(N^3)$. Let define $\mathbf{D}=\mathbf{A}^{-1}$. Now, assume that $\...

**3**

votes

**0**answers

55 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),\...

**2**

votes

**0**answers

63 views

### Bound for the inverse of a summation of rank-1 matrices

Given vectors $x_1,\dots,x_T \in R^d$ satisfying $\|x_i\|_2 = 1$, define $A_0 = I$ and $A_t = I + \sum_{i=1}^tx_ix_i^\top$ for $t \geq 1$. We are interested in the following quantity:
\begin{align}
S_{...

**1**

vote

**0**answers

89 views

### Is this matrix invertible?

Suppose that $P$ is a finite partition of the unit $d$-dimensional hypercube composed of $m$ hyperrectangles $R_1,...,R_m$
Denote $u_1,...,u_m$ the middle-point of each hyperrectangle : $u_i = \frac{...

**0**

votes

**0**answers

59 views

### Unboundedness of pseudo-inverse

Let $H$ be a real Hilbert space, and let $T$ be a bounded self-adjoint operator of $H$ to itself. Let $T^+$ denote the pseudo-inverse operator of $T$, that is the operator defined to be zero on $(TH)^{...

**7**

votes

**1**answer

656 views

### What are known properties of matrices where off-diagonal elements are 1?

Consider a matrix where the diagonal entries are anything but the off-diagonal entries are all one. I was able to find a formula for the determinant of this matrix, but what are other known properties?...

**3**

votes

**0**answers

43 views

### Solutions to a special confluent Vandermonde system

Consider the polynomial $P(X) =\prod_{i=1}^k (X-x_i)^s$ and let $M$ be the corresponding confluent Vandermonde matrix. Concretely, here is what I mean by that. Define
$$
M^{(0)} = \begin{pmatrix}
1 &...

**3**

votes

**2**answers

293 views

### Inequality for 0-1 matrices

Given an $n \times n$-matrix $A$ with entries only $0$ or $1$ and determinant equal to $\pm 1$. Define the magnitude $M_A$ of $A$ as the sum of all entries of the inverse of $A$.
Question 0: Do we ...

**1**

vote

**1**answer

158 views

### A closed-form expression for the inverse of a block-matrix

Let $\bf A$ be an $n \times n$ non-singular matrix over $\mathbb{F}$.
Let $x$ be a non-zero element of $\mathbb{F}$.
Suppose that ${\bf 1}_{n}$ is a symbol for the all-one vector of length $n$ ...

**3**

votes

**0**answers

48 views

### Singularity of symmetric block matrix with singular diagonal blocks

One can show that the following statement holds:
Given symmetric matrix $A \in \Re^{n \times n}$ and tall matrix $B \in \Re^{n \times p}$ with full column rank,
$$\begin{bmatrix}A & B \\ B^T &...

**2**

votes

**1**answer

159 views

### How to derive the mean of inverse-Wishart distribution?

How to derive the mean of inverse-Wishart distribution in
Inverse-Wishart distribution?
I have no idea about it. Thanks for your help.

**1**

vote

**0**answers

55 views

### What is the expected inverse of 1 plus a Wishart distribution?

Let $$ X \sim \mathcal{W}_{n} (n, \Sigma) \; \;$$
Where $\mathcal{W}$ denotes the Wishart distribution and $\Sigma \in \mathbb{R}^{q \times q}$ is the corresponding scale matrix (symmetric, positive ...

**1**

vote

**0**answers

57 views

### Direct solution to maximum likelihood computation problem using the derivative of multivariate Gaussian w.r.t. covariance matrix

For an application, I need to compute the maximum loglikelihood of data coming from a $d$-dimensional multivariate Gaussian random variable:
$$
\textbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)
$$
...

**3**

votes

**1**answer

63 views

### Difference of pseudoinverse bound under assumptions

This seems like a standard problem, but unable to find a solution online.
Suppose we have two singular PSD matrices A and B with the following assumptions:
$ 0 < x \leq ||A|| \leq y$
$ 0 < ||...

**1**

vote

**2**answers

76 views

### Controllability Gramian asymptotics for small times

Set-up. Consider the following linear controlled system
$$
\dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1)
$$
where $y$ is the state of the system, $y(t) \in R ^n$, $A \in R ...

**5**

votes

**1**answer

231 views

### Trace of inverse of random positive-definite matrix in high dimension?

Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...

**1**

vote

**0**answers

83 views

### Angle inequality for inverse of PD diagonal matrix

I have two diagonal, positive definite matrices $A$ and $B$, and I have that the angle between $Ax$ and $x$ is smaller than that of $Bx$ and $x$:
$$\frac{x^T A x}{\|Ax\|_2} \geq \frac{x^T B x}{\|Bx\|...

**1**

vote

**0**answers

22 views

### Explicit Formula for the Pseudo Inverses of a Family of Matrices Related to Calculating Certain Vertex Weights

Trying to identify canonical vertex weights for complete weighted graphs, I investigated the Ansatz, that the sum over the weights of all edges that are adjacent to same vertex, should equal the sum ...

**0**

votes

**0**answers

197 views

### How to solve optimization with matrix pseudo inverse

Thank everybody in advance. I'd like to solve an optimization problem for a matrix function $f(C)$. However, the matrix pseudo-inverse constraint gives me big troubles.
Vectors $V_{n\times 1}$, $F_{m\...

**6**

votes

**1**answer

319 views

### $(AB)^+\approx B^+A^+$ for $B$ “fat” enough?

Let $A\in\mathbb{R}^{r\times m}$ be a matrix of full row rank, and let $\cdot^+$ denote the Moore-Penrose inverse.
Consider a sequence of matrices $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...

**7**

votes

**1**answer

816 views

### Determinant and Inverse of a Toeplitz matrix

Let $T(n,k)$ be a $n \times n$ symmetric Toeplitz matrix, where all the entries of first $k$ super-diagonal (and sub-diagonal), last $k-1$ super-diagonal (and sub-diagonal) are ones, and rest of the ...

**0**

votes

**1**answer

80 views

### How to derive the solution of Tikhonov Regularization via SVD [closed]

The solution to Tikhonov Regularization is
$$x=(A^HA+\sigma^2_{min}I)^{-1}A^Hb$$
where $\sigma^2_{min}$ is the minimum of the singular values of $A$.
Then we apply $SVD$ to $A$ such that,
$$A=U\Sigma ...

**1**

vote

**0**answers

141 views

### Derivative of complex matrix pseudo inverse with respect to real and imaginary components

I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$.
I am interested in evaluating the ...

**2**

votes

**3**answers

309 views

### Inverse of matrix $D + ADA^T$

Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently?
$$(D + ADA^T)^{-1}$$
Hints or references are ...

**3**

votes

**1**answer

481 views

### Approximating the expectation of a matrix inverse

Let
$$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$
where $A$ is a given $n \times m$ matrix (where $m \gg n$),
$$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$
...

**11**

votes

**2**answers

884 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 ...

**2**

votes

**1**answer

595 views

### Inversion of the sum of an identity matrix and two Kronecker products

Writing the SVDs of symmetric positive-definite matrices $A=U_AD_AU_A^T$ and $B=U_BD_BU_B^T$, the following inversion can conveniently be simplified:
\begin{align*}
(I + A \otimes B)^{-1}
&= ((U_A ...

**2**

votes

**2**answers

2k views

### Inverse of particular lower triangular matrix

I have an $n \times n$ lower triangular matrix $A$ where
$$A_{i,j} = {\bf x}_i{\bf x}_j^H,\quad i>j$$
$$A_{ii}=1, \quad 1 \leq i \leq n,$$
and ${\bf x}_i$ is a $1 \times k$ (row) vector, where $...

**3**

votes

**0**answers

154 views

### The inverse of sum of two positive matrices with almost orthogonal supports

I am interested to find an approximate formula for
$$A (A+B)^{-1} A\ ,$$
for two positive matrices $A$ and $B$ whose supports are almost orthogonal.
If the support of $A$ and $B$ are orthogonal ...

**0**

votes

**1**answer

237 views

### What is the most accurate and efficient method of finding an inverse of a hessian matrix?

For any hessian matrix, of say 300 by 300, and may or may not necessarily be positive semi-definite, thus algorithms such as Cholesky decomposition may not be used.
I've found that some algorithms ...

**11**

votes

**0**answers

263 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,...

**9**

votes

**2**answers

658 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)\...

**3**

votes

**1**answer

190 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 ...

**2**

votes

**1**answer

418 views

### When is the following block matrix invertible?

Let
$$A = \begin{bmatrix}
x_{11} A_{11} & x_{12} A_{12} & x_{13} A_{13} & \cdots & x_{1d} A_{1d}\\
x_{21} A_{21} & x_{22} A_{22} & x_{23} A_{23} & \cdots & x_{2d} ...

**9**

votes

**1**answer

338 views

### Inverse of a matrix with binomial entries

This is closely related to this question: Eigenvalues of a matrix with binomial entries.
We consider the matrix:
$$M_{ij} = 4^{-j}\binom{2j}{i}$$
where it is understood that the binomial ...

**5**

votes

**1**answer

448 views

### Invertibility of the Schur Complement

Suppose that
$$
M = \begin{bmatrix}A & B \\ C & D\end{bmatrix}.
$$
I know that if $D$ and $M\setminus D$ (where $M\setminus D$ is the Schur Complement of $D$ in $M$) are invertible, then $M$...

**3**

votes

**2**answers

652 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 ...

**5**

votes

**1**answer

123 views

### Find the inverse of a more general matrix that is similar to the Hilbert matrix

In the last MO question , the following matrix is given:
$$M_{ij}=\left[\frac{1+(-1)^{i+j}}{i+j-1}\right]$$
and its inverse has been discussed.
Now the problem is further extended to a more general ...

**10**

votes

**3**answers

667 views

### Find the inverse of a matrix that is very similar to the Hilbert matrix

The standard Hilbert matrix $H$ is given by
$$H_{ij}=\frac{1}{i+j-1},$$
and it has an inverse given for example in this MO question.
Now I have encountered a matrix $M$ of similar form, namely,
$$...

**5**

votes

**2**answers

253 views

### minimum-maximum entries matrix

Let $M(n)$ be an $n\times n$ matrix in the variables $x_1,\dots,x_n$ with entries
$$M_{i,j}(n)=\frac{x_{\max(i,j)}}{x_{\min(i,j)}}, \qquad 1\leq i,j\leq n.$$
I'm interested in the following:
...

**0**

votes

**1**answer

1k views

### How to calculate the inverse of the sum of kronecker products with the identity matrix

How to calculate $G^{-1}$ efficiently when $G$ is a large matrix knowing that:
\begin{eqnarray}
G=I⊗A + A⊗I
\end{eqnarray}
Or since i'm using $G^{-1}$ to multiply by some other matrix, how to find $...

**2**

votes

**0**answers

124 views

### Left multiplication Fréchet Differentiable

Let $L^2$ be a separable Hilbert space and let $\{e_i(t)\}_{i \in \mathbb{N}}$ be a basis for it. Moreover let $\phi(T,z)$ and $\psi(T,z)$ be 1-parameter families (in z) in $L^2$, with basis ...

**3**

votes

**1**answer

164 views

### Are inverse eigenvalue problems (IEPs) hopeless and not a fruitful area of research?

I've been studying IEPs, in particular, the Nonnegative Inverse Eigenvalue Problem, some basic theoretical framework, the many open questions that IEPs have, and now sort of realize the computational ...

**3**

votes

**1**answer

1k views

### Transforming a non-invertible matrix into an invertible matrix

Given a non-invertible, diagonalizable matrix $A$, I wish to transform it into another matrix $B$ that satisfies:
$B$ is invertible
all non-zero eigenvalues of $A$, are also eigenvalues of $B$
all of ...

**1**

vote

**2**answers

745 views

### Is there an easy way to find the inverse of a symmetric block matrix with off-diagonal rank-one matrices?

The symmetric matrix I need to invert is of the following form:
\begin{align}
J_e=\left(\begin{matrix}-I&B_{11}&B_{12}&...&...&B_{1(N-1)}\\
B_{11}&-I&B_{22}&....

**1**

vote

**1**answer

147 views

### Find a matrix and its inverse satisfying lower and upper bounds

I reduced a problem of matrix completion to the problem
find $A,B$ such that
$AB=I$
$A_{min}\leq A \leq A_{max}$
$B_{min}\leq B \leq B_{max}$
One possible approach would be to just minimize $\|AB-...

**1**

vote

**0**answers

119 views

### Computing Moore-Penrose generalized inverse using convergent geometric series

I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations.
Let $A$ be an $...