For questions about inverses and pseudoinverses of matrices.

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0answers
10 views

Stucture of inverse (MP) of totally positive rectangular matrix

The special structure of inverse of non-singular totally positive square matrix (whose all entries are positive) discussed in MO(see here). The inverse has a special structure (M-matrix). With some ...
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0answers
22 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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0answers
43 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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1answer
92 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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1answer
80 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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1answer
60 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 ...
4
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2answers
162 views

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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1answer
257 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 ...
0
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1answer
51 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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1answer
93 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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1answer
95 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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2answers
195 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 ...
3
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0answers
983 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} ...
6
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2answers
660 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 ...
0
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0answers
100 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 ...
4
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0answers
114 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} ...
1
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0answers
886 views

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 ...
2
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1answer
459 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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2answers
361 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 ...
3
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1answer
487 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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1answer
201 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 ...
2
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0answers
170 views

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 ...
5
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3answers
943 views

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 ...
2
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0answers
480 views

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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1answer
116 views

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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1answer
134 views

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
votes
1answer
497 views

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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0answers
127 views

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
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0answers
579 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 ...
3
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1answer
649 views

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
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1answer
314 views

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 ...
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1answer
5k views

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 ...
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2answers
246 views

On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: http://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 ...
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0answers
132 views

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 ...
5
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0answers
688 views

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
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0answers
338 views

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 ...
5
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1answer
4k views

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} ...
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4answers
339 views

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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1answer
385 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 ...