Questions tagged [matrix-inverse]
For questions about inverses and pseudoinverses of matrices.
137
questions
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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 $\...
1
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1
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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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2
answers
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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:
$$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} &...
1
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2
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727
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Inverse of a matrix expression
Let
$$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 ...
8
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0
answers
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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{...
8
votes
2
answers
2k
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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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160
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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 ...
4
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0
answers
377
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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}
\...
4
votes
1
answer
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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 ...
4
votes
1
answer
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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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2
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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 ...
3
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1
answer
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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 &...
0
votes
1
answer
673
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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 ...
2
votes
0
answers
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Inverting a matrix with entries equal to positive or negative infinity
I would like to define an inverse on matrices whose entries may be positive or negative infinity.
To formulate my problem precisely, suppose that I have a matrix $A$ and another matrix $B$. How do I ...
18
votes
4
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Sherman-Morrison type formula for Moore-Penrose pseudoinverse
Given an $n\times n$ invertible matrix $\mathbf A$ and two column vectors $\mathbf u$, $\mathbf v\in\mathbb R^n$, suppose that $1 + {\mathbf v}^T {\mathbf A}^{-1}\mathbf u \neq 0$.
Then the Sherman-...
2
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0
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Update the inverse of sum of two symmetric matrices
There are two invertible symmetric matrices A and B, of which B is a block diagonal. A and B have the same dimensions. I need to iteratively calculate the inverse of M = s * A + B, where s is a ...
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1
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How to calculate $(A^{-1})_{ii}$ for an invertible hyperhermitian quaternionic matrix $A$?
The article
Alesker, S. (2003). Quaternionic Monge-Ampere equations. The Journal of Geometric Analysis, 13(2), 205-238.
has the following CLAIM:
Claim. Let $A$ be an invertible hyperhermitian ...
1
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1
answer
155
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Do there exist a way to solve inhomogeneous matrix equations Ax = b for only selected rows?
The inhomogeneous matrix equation $\mathbf{A} x = b$ can be solve in many ways, but in this particular case, I am looking for a solution to this problem on a set of constraints.
The matrix $A$ is ...
6
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1
answer
979
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Best rank one approximation
Assume $u,v\in\mathbb{C}^n$ are complex vectors. I was wondering if there is a closed form expression for the following problem in terms of $u$ and $v$
\begin{equation*}
\arg\min_{x\in\mathbb{C}^n} \|...
1
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0
answers
221
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positiveness of the inverse solution to Sylvester equation
I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form:
$$
\mathbf{M} = \begin{vmatrix}
\mathbf{A} & \mathbf{b} \\\
\...
0
votes
0
answers
945
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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 ...
3
votes
1
answer
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Expected value of trace of matrix inverse
Given a $N\times K$ matrix $A$ of full rank with $ K < N $, a diagonal matrix $D$ and knowing that $E[D]=bI_N$, where $E[\cdot]$ is the expected value and $I_N$ is the $N\times N$ identity matrix ...
0
votes
1
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494
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Calculate the inverse of a matrix
Hi
I have a equation $$Ax=b,$$ where matrix $A$ is invertible, $b$ is a constant vector, and $x$ is the unknown vector. To obtain $x$, it is obvious $x=A^{-1}b$. Alternatively, if $A$ is Hurwitz, one ...
1
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1
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Derivative of log determinant and inverse
I have a matrix $\Sigma$ with element $(i,j)$
$$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$
The matrix is positive definite and symmetric (it is a covariance matrix).
Now I need to evaluate
$$\frac{\...
8
votes
2
answers
700
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On the convexity of element-wise norm 1 of the inverse
Question first asked on math.stackexchange here: https://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse
On the convexity of element-wise norm 1 of the ...
17
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1
answer
4k
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Geometric interpretations of matrix inverses
$A$ is an invertible $n \times n$ matrix. Interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point (...
1
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0
answers
195
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bounds on the entries of an inverse circulant matrix
Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the ...
6
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0
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When is the inverse diagonally dominant?
There is a large literature devoted to studying the inverses of diagonally dominant matrices. I'd like to know if there is information about a so-to-say opposite situation: we have a matrix $A$ and ...
2
votes
0
answers
350
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Fast inversion of a special kind of matrices - approximations are ok
Suppose I have a stochastic matrix $M$ (with thousands or millions of stochastic column vectors), which I split into two matrices: $D$ containing only the diagonal entries of $M$, and $R$ containing ...
11
votes
2
answers
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How to calculate the inverse of the sum of an identity and a Kronecker product efficiently?
I have a matrix $K$ which is the sum of a identity and a Kronecker product of two symmetric matrices as following and I want to calculate the inverse of it $K^{-1}$.
\begin{eqnarray}
K=\mathbf{I}_{mn}+...
2
votes
4
answers
604
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inverse-closed matrix spaces
Is there a known characterization of such spaces?
An example: the space of $n \times n$ matrices spanned by $I$ and $J$ (the identity and all-ones matrices, respectively) is inverse closed by the ...
1
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1
answer
852
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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 ...
126
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13
answers
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Should the formula for the inverse of a 2x2 matrix be obvious?
As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
2
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1
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Inverse of an AR(1) or Laplacian (?) or Kac-Murdock-Szegö matrix
My current problem involves having an exact (symbolic) inverse of a scaled AR(1) matrix for $n$-dimension. (I don't know what this matrix would be called in general; I'm sure it is used often.) This ...
15
votes
3
answers
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Questions on Toeplitz matrices: invertibility, determinant, positive-definiteness
These questions are probably very basic but I'll dare to ask them anyway since I didn't have much luck in Math Stack Exchange.
Let $A$ be an $n \times n$ Hermitian Toeplitz matrix:
$$A = \begin{...
2
votes
2
answers
4k
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Moore-Penrose pseudo inverse
I have an $n\times p$ matrix $Z$ with $p>n$
I have $A$, a diagonal matrix with positive entries
I would like to know if there is a known relation (as a function of $A$) between
the Moore-Penrose ...
26
votes
6
answers
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Deriving inverse of Hilbert matrix
The Hilbert matrix is the square matrix given by
$$H_{ij}=\frac{1}{i+j-1}$$
Wikipedia states that its inverse is given by
$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-...