Questions tagged [matrix-inverse]
For questions about inverses and pseudoinverses of matrices.
137
questions
3
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0
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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
$$
\operatorname*{rank}G(x)= \operatorname*{rank}G(y),\...
2
votes
0
answers
266
views
Bound for the inverse of a summation of rank-1 matrices
Given vectors $x_1,\dots,x_T \in R^d$ satisfying $\|x_i\|_2 = 1$, define $A_0 = I$ and $A_t = I + \sum_{i=1}^tx_ix_i^\top$ for $t \geq 1$. We are interested in the following quantity:
\begin{align}
S_{...
1
vote
0
answers
100
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Is this matrix invertible?
Suppose that $P$ is a finite partition of the unit $d$-dimensional hypercube composed of $m$ hyperrectangles $R_1,...,R_m$
Denote $u_1,...,u_m$ the middle-point of each hyperrectangle : $u_i = \frac{...
7
votes
1
answer
3k
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What are known properties of matrices where off-diagonal elements are 1?
Consider a matrix where the diagonal entries are anything but the off-diagonal entries are all one. I was able to find a formula for the determinant of this matrix, but what are other known properties?...
3
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0
answers
72
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Solutions to a special confluent Vandermonde system
Consider the polynomial $P(X) =\prod_{i=1}^k (X-x_i)^s$ and let $M$ be the corresponding confluent Vandermonde matrix. Concretely, here is what I mean by that. Define
$$
M^{(0)} = \begin{pmatrix}
1 &...
3
votes
2
answers
365
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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
481
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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
224
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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,
$$\begin{bmatrix}A & B \\ B^T &...
2
votes
2
answers
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How to derive the mean of inverse-Wishart distribution?
How to derive the mean of inverse-Wishart distribution in
Inverse-Wishart distribution?
I have no idea about it. Thanks for your help.
1
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0
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86
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What is the expected inverse of 1 plus a Wishart distribution?
Let $$ X \sim \mathcal{W}_{n} (n, \Sigma) \; \;$$
Where $\mathcal{W}$ denotes the Wishart distribution and $\Sigma \in \mathbb{R}^{q \times q}$ is the corresponding scale matrix (symmetric, positive ...
1
vote
0
answers
271
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Direct solution to maximum likelihood computation problem using the derivative of multivariate Gaussian w.r.t. covariance matrix
For an application, I need to compute the maximum loglikelihood of data coming from a $d$-dimensional multivariate Gaussian random variable:
$$
\textbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)
$$
...
3
votes
1
answer
378
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Difference of pseudoinverse bound under assumptions
This seems like a standard problem, but unable to find a solution online.
Suppose we have two singular PSD matrices A and B with the following assumptions:
$ 0 < x \leq ||A|| \leq y$
$ 0 < ||...
1
vote
2
answers
140
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Controllability Gramian asymptotics for small times
Set-up. Consider the following linear controlled system
$$
\dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1)
$$
where $y$ is the state of the system, $y(t) \in R ^n$, $A \in R ...
7
votes
1
answer
779
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Trace of inverse of random positive-definite matrix in high dimension?
Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
1
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0
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104
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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 $x$:
$$\frac{x^T A x}{\|Ax\|_2} \geq \frac{x^T B x}{\|Bx\|...
1
vote
0
answers
31
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Explicit Formula for the Pseudo Inverses of a Family of Matrices Related to Calculating Certain Vertex Weights
Trying to identify canonical vertex weights for complete weighted graphs, I investigated the Ansatz, that the sum over the weights of all edges that are adjacent to same vertex, should equal the sum ...
0
votes
0
answers
657
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How to solve optimization with matrix pseudo inverse
Thank everybody in advance. I'd like to solve an optimization problem for a matrix function $f(C)$. However, the matrix pseudo-inverse constraint gives me big troubles.
Vectors $V_{n\times 1}$, $F_{m\...
5
votes
1
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362
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$(AB)^+\approx B^+A^+$ for $B$ "fat" enough?
Let $A\in\mathbb{R}^{r\times m}$ be a matrix of full row rank, and let $\cdot^+$ denote the Moore-Penrose inverse.
Consider a sequence of matrices $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...
7
votes
1
answer
2k
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Determinant and Inverse of a Toeplitz matrix
Let $T(n,k)$ be a $n \times n$ symmetric Toeplitz matrix, where all the entries of first $k$ super-diagonal (and sub-diagonal), last $k-1$ super-diagonal (and sub-diagonal) are ones, and rest of the ...
0
votes
1
answer
277
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How to derive the solution of Tikhonov Regularization via SVD [closed]
The solution to Tikhonov Regularization is
$$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
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0
answers
301
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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
525
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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 ...
4
votes
1
answer
2k
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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
2k
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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
1k
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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 ...
3
votes
2
answers
3k
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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
267
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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
759
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
303
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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
$$
A^{-1}(k,...
9
votes
2
answers
874
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Inverse of special upper triangular matrix
Consider the following $n \times n$ upper triangular matrix with a particularly nice structure:
\begin{equation}\mathbf{P} = \begin{pmatrix}
1 & \beta & \alpha+\beta & \dots & (n-3)\...
3
votes
1
answer
321
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Inverting (via Taylor expansion) a sum of (rank-deficient) skew-symmetric matrix and (rank-deficient) Diagonal matrix
I have the following problem:
A matrix $C\in \mathbb{R}^{2N}$, where
$C=\epsilon A+D$
$\epsilon A=(C-C')/2$ is skew symmetric with "block" anti-diagonal structure of size 4.
$ D=(C+C')/2$ (Diagonal ...
2
votes
2
answers
991
views
When is the following block matrix invertible?
Let
$$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
580
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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 ...
6
votes
1
answer
1k
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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 $M$...
4
votes
2
answers
1k
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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
193
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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
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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,
$$...
5
votes
2
answers
304
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minimum-maximum entries matrix
Let $M(n)$ be an $n\times n$ matrix in the variables $x_1,\dots,x_n$ with entries
$$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
2k
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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
126
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 ...
5
votes
1
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267
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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
2k
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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
1k
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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
165
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
138
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 $...
1
vote
1
answer
168
views
Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $t = \pm 1$, where $\psi_k(t) = t^k$?
Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})^\top$ - a polynomial basis.
Suppose there is a matrix
$$
A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] \...
1
vote
1
answer
2k
views
Largest element in inverse of a positive definite symmetric matrix [closed]
If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
2
votes
1
answer
11k
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Computing the inverse of a Cholesky decomposition [closed]
I have chol(A) and I would like chol(A^-1). One way to do this is to construct the inverse positive definite symmetric matrix and then take its Cholesky decomposition (with Dpotri and Dpotrf for ...
7
votes
2
answers
11k
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Relation between eigenvalues of $A$ and $A^TA$?
For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$?
I ask this because I am looking into the relation between $A$ and $A+cI$, ...
8
votes
1
answer
1k
views
Efficiently compute the trace of a sparse matrix times the inverse of a sparse matrix?
How can I efficiently compute
$\mathrm{trace}(A(B^{-1}))$
where $A$ and $B$ are both sparse symmetric PSD $n \times n$ matrices, both with $O(n)$ non-zero entries? If it helps, the pattern of non-...