# Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

129
questions

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### Implicit function theorem / Implicit selections when Jacobian not invertible

I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...

2
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0
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101
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### Decompose a rational matrix as an integer matrix and an inverse of integer matrix

Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...

2
votes

1
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51
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### Product of a vector by an inverse of Toeplitz matrix

It is well known that using fast Fourier transform it's possible to multiply a vector by a Toeplitz matrix $A \cdot v = w$ in $n\cdot\log(n)$ operations.
I read somewhere that also the product of a ...

3
votes

1
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249
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### Inverting a matrix using the Matrix logarithm

This is probably going to lead nowhere, but maybe it be possible to use the matrix logarithm to invert matrices?
For positive definite matrices, we have that the logarithm exists and
$$
\log(A^{-1})= -...

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0
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118
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### Infinite dimensional matrix solvability

In order to solve a boundary problem for a conducting hemisphere, the following matrix equation arises (derived from the boundary condition on the curved part of the surface) where we must solve for ...

0
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### Given optimality of L1 norm, prove that absolute value of sum of a vector with proper sign is less than 1?

Problem:
Given a domain $\mathcal{D}\subset\mathbb{R}^{l}$, we can find $l$
points $\boldsymbol{v}_{i}\in\mathcal{D}$, $i=1,\cdots,l$. Each
point is a column vector with dimension $l\times1$. They ...

3
votes

1
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138
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### Complexity of inverting and multiplying against a symmetric Toeplitz matrix with two repeated entries

I know that the computational complexity of inverting a general $n \times n$ matrix $A$ is $O(n^{2.373})$ and multiplying it against an $n \times m$ matrix is $O(n^2m)$. Moreover, I've seen that ...

0
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0
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71
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### Inverse of a matrix defined as a quadratic form

Suppose I have a vector $x$ of $n$ reals, and tensors $A,B$ of respective sizes $(n,k,k,n)$ and $(k,k,n)$, so that both the array-vector product $Bx$ and the quadratic form $x'Ax$ are of size $(k,k)$.
...

4
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2
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408
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### What is the inverse of a triangular matrix whose nonzero elements are binomial coefficients? What is the closed-form solution to a recursive relation?

Let
\begin{equation*}
\begin{split}
M_m
&=\begin{pmatrix}
-\binom{1}{0} & \binom{2}{0} &-\binom{3}{0} &\dotsm & (-1)^{m-1}\binom{m-1}{0} & (-1)^m\binom{m}{0}\\
0 & \binom{2}...

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0
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75
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### Algorithm for efficiently calculating $(A+\sum_{i=1}^n B_i)^{-1}$ where $A^{-1}\in\mathbb S^n_+$ is known and $B_i$ are sparse matrices

Let $A\in\mathbb R^{n\times n}$ be a symmetric positive-definite matrix and $A^{-1}$ is already known. Now I want to compute the matrix $(A+\sum_{i=1}^n B_i)^{-1}$ where each $B_i$ is a sparse ...

2
votes

2
answers

105
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### Orthonormal solution of overdetermined linear equations

I have a two matrices $A$ and $B$ in $\mathbb{R}^{m \times n }$ ($m \gg $ n) such that there exists an orthonormal matrix $X \in \mathbb{R}^{n \times n }$, such that:
$$AX = B$$
Given that $X$ is ...

2
votes

2
answers

220
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### Full-rank matrix

I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem).
$$\left[\begin{array}{cccccccccc}
0 & 1 & 1 & 1 & 0 & 0 & ...

0
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1
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88
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### Approximating the expectation of trace inverse of random Gaussian combination

Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. Has anyone studied this expectation in asymptotics $$E_{A}(\mathrm{Tr}( (A^T ...

2
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1
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171
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### Trace inverse of random PSD matrix?

Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. I am interested in the expectation of $$E_{A}(\mathrm{Tr}( (A^T A + \lambda \...

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1
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160
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### Proving 2 matrices have the same trace [closed]

I found a problem in my textbook and I have tried solving it, but I had no succes. The problem is:
Let $A$ and $B$ be $n \times n$ matrices with complex number entries. Given that $AB−BA$ is ...

1
vote

1
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33
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### Characterization of pseudo unit-matrices

I accidently found a class of rank-deficient square matrices that are their own Moore-Penrose pseudo inverse:
$$\boldsymbol{A}\in\mathbb{R}^{n\times n}: a_{(i,i)}=n-1,\, a_{\lbrace i,j\rbrace}=-1\...

2
votes

1
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147
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### Fast inverse of asymmetric diagonally dominant matrix with diagonal 1 and non-positive off-diagonals

I am interested in ways to obtain (even approximately) the inverse of an asymmetric diagonally dominant matrix with diagonal 1 and non-positive off-diagonals.
Formally, let $A$ be a $n\times n$ matrix ...

1
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1
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242
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### Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible

$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix:
$$
M:= \begin{bmatrix}
\cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\
\cPr{X=...

0
votes

1
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137
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### Does there exist a function $f(X)$ with the following gradient $\mathrm{Tr}[(I-X)^{-1}]\cdot g(X)$?

Let $f: \mathbb{R}^{n\times n} \to \mathbb{R}$ be a function that receives a square matrix and spits out a scalar. Does there exist a function $f$ such that the gradient $\nabla_Xf(X) = \mathrm{Tr}[(...

0
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1
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127
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### Product of matrices equal identity

I need to solve the following equation for the matrix $P \in\mathbb{R}^{r\times d}:$
$$
((PAP^\top)^{-1} P S P^\top (PAP^\top)^{-1})^2 = I_r,
$$
where $S$ is a symmetric $d\times d$ matrix, $A$ is a ...

2
votes

2
answers

227
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### When is $(I - X)^{-\top} \circ X = 0$?

I am currently looking at the following expression $(I - X)^{-\top} \circ X = 0$, where $\circ$ is the Hadamard product, $\top$ is the transpose, $I$ is the identity, and $X$ is non-negative and ...

2
votes

1
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134
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### Existence of matrices with some invertibility properties

Prove that there exists five matrices $B_i \in \mathbb{F}_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B_i$'s form an invertible matrix in $\mathbb{F}_2^{10\times 10}$.
I am interested ...

3
votes

1
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334
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### Minimum upper bound for sum of the entries of the inverse covariance matrix

Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel
$$k(x, x') := \exp \left(-\frac{\|x-x'\|^2}{2\sigma^2}\right)$$
and let $\mathbf{K}$ be the following $n \times n$ covariance matrix
$$\mathbf{K} = \...

1
vote

1
answer

147
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### If $\Vert A-B\Vert_{op}\leq \varepsilon$ then $A^{-1}$ and $B^{-1}$ are uniformly close

Let $A,B$ are two $p\times p$ positive definite matrices such that $0<\delta_0\leq \min\{\lambda_{\min}(A), \lambda_{\min}(B)\}\leq \max\{\lambda_{\max}(A), \lambda_{\max}(B)\}\leq \delta_1$. Also ...

4
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2
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462
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### Do any two hermitian matrices A and B commute with the support of their commutator?

Let $A$ and $B$ be Hermitian matrices. Let $[A,B] = AB - BA$ be their commutator and let $[A,B]^+$ be the Moore-Penrose pseudoinverse of $[A,B]$.
Is it then true that $A$ and $B$ both commute with the ...

2
votes

1
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403
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### Mistake in Karl Pearson's 1900 paper introducing the chi-squared distribution

Background
I'm reading Karl Pearson's 1900 paper titled:
On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be ...

2
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1
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557
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### Matrix derivative w.r.t. a general inverse form: $(A^TA)^{-1/2}D(A^TA)^{-1/2}$

I want to find derivative of matrix $(A^TA)^{-1/2}D(A^TA)^{-1/2}$ w.r.t. $A_{ij}$ where D is a diagonal matrix. Alternatively, it is okay too to have
$$\frac{\partial}{\partial A_{ij}} a^T(A^TA)^{-1/2}...

0
votes

2
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102
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### Computation to differentiate a determinant [closed]

Consider a positive Hermitian $N \times N$ matrix $A$ with complex valued coefficients. We list its eigenvalues in increasing order and with their multiplicities, $\mu_{1} \leq \mu_{2} \leq \cdots \...

2
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1
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302
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### Ratios of Gaussian integrals with a positive semidefinite matrix

Cross-post from MSE
https://math.stackexchange.com/questions/4118128/ratios-of-gaussian-integrals-with-a-positive-semidefinite-matrix
Generally speaking, I’m wondering what the usual identities for ...

4
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740
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### Sufficient conditions for invertibility of a block tridiagonal matrix

Let $M_n \in \mathbb{R}^{N \times N}$ be a block-tridiagonal matrix:
$$M_n = \begin{bmatrix} B_1 & C_1 & 0 & 0 & \cdots & 0 \\ A_1 & B_2 & C_2 & 0 & \cdots & 0 \...

2
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1
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146
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### Closed form for the inverse of a special transition matrix

Let $b\in\mathbf{R}^n$ be fixed, and let $A\in\mathcal{M}_{n\times n}(\mathbf{R})$ be given but otherwise arbitrary. Let $a_1,\ldots,a_n$ denote the coefficients of the characteristic polynomial of $A$...

0
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1
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120
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### Expected pseudo-inverse of isotropic random matrix

Suppose I have a random $m \times m$ matrix $R \sim \mu$ that is possibly singular. Is it true that
$ E[R] \propto I$ implies that there exists a scalar $r_{\mu, m}$ such that $E[R^+] = r_{\mu, m}I$, ...

4
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1
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462
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### Condition number for matrix of eigenvectors of a diagonalizable matrix

Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$.
For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} ...

2
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1
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522
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### Matrix inversion inequality

Suppose $A, B, C\in\mathbb{R}^{n\times n}$ are all symmetric positive definite matrices, and they satisfy the inequality $A \succeq B + C$. Assume also that all of the three matrices are bounded, i.e.,...

2
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1
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485
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### Complexity of pseudoinverse of a low-rank matrix

I have a large matrix $\mathbf{A}\in \mathbb{C}^{m\times n}$ with very low rank $r$, I find that the general complexity of finding its pseudoinverse is $\mathcal{O}(\max(m,n)^3)$, this is too high. ...

32
votes

1
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### Inverting lower triangular matrix in time $n^2$

I have a lower $n\times n$ triangular matrix called $A$ and I want to get $A^{-1}$ solved in $O(n^2)$. How can I do it?
I tried using a method called "forward substitution", but the ...

6
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3
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840
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### How to calculate inverse of sum of two Kronecker products with specific form efficiently?

I have a matrix with specific form of $A\otimes I + B\otimes J$ where $A$ and $B$ are general dense matrices, $n\times n$. $I$ is an $m\times m$ identity matrix. $J$ is a $m \times m$ dense matrix ...

1
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0
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146
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### Random matrix invertible

I am trying to figure out why the following random matrix is invertible:
\begin{align*}
A_j = I_d + J_{\mu}(\hat{X}_{t_{j-1}})(t_j - t_{j-1}) +
\left( \begin{array}{rrr}
B_1^T \\ ...

5
votes

1
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373
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### Inverse of a Cauchy-like matrix

Consider $n\times n$ symmetric Cauchy-like matrix $M$ with elements $(M_{ij})_{i,j=1}^{n}$ given by
$$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} \frac{...

1
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0
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76
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### 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, ...

2
votes

1
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787
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### 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 & ...

1
vote

1
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172
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### 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

70
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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

238
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### 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
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0
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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

2k
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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
votes

0
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70
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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

360
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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

459
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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

200
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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 &...