# Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

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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\...

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278 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 $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...

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209 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 ...

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53 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 ...

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77 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 ...

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278 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 ...

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221 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 $m \gg n$),
$$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$
...

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54 views

### Calculation of $\chi^2$ for very small covariance matrices

I produced simulations of data for my experiments, changing each time my initial parameter $\theta$ , $n$ times: $\theta_1\, \theta_2, \cdots \theta_N$. For each realisation of my data I calculated ...

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546 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 ...

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231 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 ...

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539 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 $...

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116 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 ...

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124 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 ...

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250 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,...

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511 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)\...

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489 views

### Invertible Markov matrices

Are there papers or books which reference any necessary or sufficient conditions on which a Markov matrix (a stochastic matrix) is invertible?

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135 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 ...

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222 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} ...

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227 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 ...

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181 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 $M$...

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419 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 ...

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107 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 ...

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622 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,
$$...

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245 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:
...

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622 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 $...

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123 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 ...

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156 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 ...

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662 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 ...

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561 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}&....

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137 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-...

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94 views

### Inverse of a correlation matrix that has arcsin elements

Let $C$ be a correlation matrix whose off-diagonal elements are defined as follows: $C(i,j)=arcsin(\sqrt{k^{|i-j|}}$ where $k<1$. Can the inverse of the correlation matrix be derived analytically?

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100 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 $...

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126 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] \...

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588 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 ...

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3k views

### 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 ...

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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 $A+cI$, ...

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558 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-...

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158 views

### Approximate $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$ by Taylor expansion [closed]

Suppose we have a complex matrix $\mathbf{M}$. Let $\mathbf{M}^+=(\mathbf{M}^*\mathbf{M})^{-1}\mathbf{M}^*$ be the pseudo-inverse of $\mathbf{M}$, where $^*$ denotes the conjugate transpose. Let $\...

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571 views

### Inverse of Matrix with one element approches infinity [closed]

Let A be non singular matrix of order N and inverse of A is known. Is it possible to find/approximate inverse of A if only one element of A; a(i,j); is replaced by number approaching infinity(M = big ...

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3k views

### Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation

Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of:
$$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} &...

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469 views

### Inverse of a matrix expression

Let
$$X_i = \left(I - P\left(I - t_it_i^T\right)\right)^{-1}$$
where $P$ is an $N\times N$ matrix and $t_i$ is a vector of $N$ elements.
Is there a way to simplify this expression in order to ...

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3k views

### Partitioned inverse 3x3 block matrix

We know that matrices can be inverted blockwise by using the following analytic inversion formula:
\begin{equation}
\begin{bmatrix} \mathbf{A} & \mathbf{C^T} \\ \mathbf{C} & \mathbf{D} \end{...

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1k views

### Partial inverse of a matrix - or does it have its own name?

In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name.
That is, a matrix (here ...

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136 views

### Applying a linear operator to a basis set following SVD orthonormalization

Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases.
My intention is to:
decompose $\Phi$ via SVD:
$U \Lambda V^T = \Phi$
to create it's ...

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195 views

### Efficiently factorize a KKT system with block diagonal upper corner

I have a system resulting from a quadratic energy minimization with linear equality constraints enforced with Lagrange multipliers which has the form:
\begin{equation}
A =
\left[\begin{array}{c|c}
\...

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2k views

### Numerical trace of inverse matrix from Cholesky

This question was somewhat answered here: Fast trace of inverse of a square matrix. However, I feel like there was no complete answer wrt the Cholesky case.
I have the matrix $\Sigma=LL^T$. Is there ...

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1k views

### optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...

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1k 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 ...

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977 views

### Calculating a generalized inverse (Moore–Penrose pseudoinverse)

I am considering a graph with $n$ edges with the following nicely structured adjacency matrix:
\begin{equation}
A_n=
\begin{pmatrix}
0 & 0 & 0 &\cdots & 0 & 0 & 1\\
0 & 0 &...

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396 views

### Perturbation of Cholesky decomposition for matrix inversion

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$
where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...