Skip to main content

Questions tagged [inverse]

Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

920 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
7 votes
0 answers
132 views

Is there a polynomial $p$ such that it is bijective and $ p: \mathbb{Q}^n \rightarrow \mathbb{Q}$ for $ n>1$ ??

Let us define a polynomial $p$ defined as follow $$p: \mathbb{Q}^n \rightarrow \mathbb{Q}.$$ I acrossed this question in mind. Is there a polynomial $p$ such that it is bijective and $p: \mathbb{Q}...
zeraoulia rafik's user avatar
6 votes
0 answers
189 views

Proving or disproving this matrix $V$ is invertible.

Here is a question on the invertibility of a special structured matrix: Notations: Let us take $n\in \mathbb{N}^*$ bins and $d\in \mathbb{N}^*$ balls. Denote the set $B = \{\alpha^1, \ldots, \alpha^m\}...
kaienfr's user avatar
  • 161
6 votes
0 answers
259 views

Invertibility (and eigen-decomposition) of complex symmetric matrix?

Consider $Z = K - i \omega S - \omega^2 M$, where: $\omega$ is a positive real number; $K$ is a real, symmetric, and positive semi-definite matrix; $M$ is a real, symmetric, and positive definite ...
user7440's user avatar
  • 832
6 votes
0 answers
468 views

computing the inversion of a matrix which is the sum of a Kronecker product and an identity matrix

I am using Gibbs sampling to compute the posterior of $\mathbf{S}_{N\times K}$ $(N>>K)$ while I should calculate a Gaussian likelihood which its covariance matrix is given as $$\mathbf{P}_{K^2\...
Dalek's user avatar
  • 47
6 votes
0 answers
724 views

Computing one-sided inverse of a matrix over some finite field

Let $M$ be a $k\times n$ matrix with $k < n$, and assume that $\text{rank}(M)=k$. Over $\mathbb{R}$, one can compute a right inverse of $M$ as follows: $$M_\text{right}^{-1} = M^T(MM^T)^{-1}$$ ...
Sadeq Dousti's user avatar
  • 3,321
6 votes
0 answers
1k views

Inverse of a product of real functions

Given $F(x) = L(x)G(x)$, with $L$ and $G$ real function strictly greater than zero. Suppose that F and G are decreasing functions (so that $F^{-1}$ and $G^{-1}$ exists). What can we say about the ...
fdesmond's user avatar
  • 131
6 votes
2 answers
348 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
royalT's user avatar
  • 71
6 votes
0 answers
177 views

Explicit quasi-inverse of Künneth-isomorphism?

With $A_X$ the complex of $\mathbb{R}$-differential forms on $X$, the Künneth theorem states that \begin{align*} A_X \otimes A_Y &\to A_{X \times Y}, \\ (\omega,\eta) &\mapsto {\rm pr}_X^\...
MaoK's user avatar
  • 117
5 votes
0 answers
370 views

Pseudoinverse as a submatrix of matrix inverse

Suppose I have a device that can compute 2x2 (complex) matrix inverses. (For now, assume only invertible matrices, $A$, are ever provided as input): $A\triangleq \begin{bmatrix} a_{11} & ...
Harry's user avatar
  • 541
5 votes
0 answers
560 views

What happens to woodbury matrix identity when A is not invertible?

The Woodbury matrix identity is \begin{equation} (A+UCV)^{-1}=A^{-1}-A^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}. \end{equation} This formula suppose that $A$, $(A+UCV)$ and $(C^{-1}+VA^{-1}U)$ are ...
G. Trav's user avatar
  • 389
5 votes
0 answers
3k views

Inverse function of hypergeometric function, e.g., ${}_{2}F_{1}(1,1;1.2;x)$

I want to know whether it is able to express the inverse function of hypergeometric function using some special function. For instance, the Gauss hypergeometric function $f(x)={}_{2}F_{1}(1,1;1.2;x)$....
Vic's user avatar
  • 463
5 votes
0 answers
1k views

Compact convergence of inverse functions

Consider two metric spaces $X$ and $Y$ and a sequence of functions $f_n\colon X\to Y$ together with a function $f\colon X\to Y$. Assume, all $f_n$ and $f$ have inverse functions $g_n$ and $g$, say. It ...
Marcel's user avatar
  • 889
5 votes
0 answers
1k views

Norm of the inverse of a tridiagonal

Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric) $$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ &...
Exodd's user avatar
  • 10.9k
5 votes
0 answers
124 views

Is there always a smooth variant of a homeomorphism between smooth manifolds?

Let $M$ and $N$ be smooth homeomorphic manifolds. Let $h:M\rightarrow N$ a homeomorphism. Does there exist $r:M\rightarrow N$ that is still a homeomorphism and additionaly smooth? Can it be chosen ...
w_w's user avatar
  • 709
5 votes
1 answer
170 views

Is the inverse of any elementary function asymptotic to some elementary function?

Is the functional inverse of any elementary function asymptotic to some elementary function ? For instance Lambert's $W(z)$ is asymptotic to $ln(z)$. See http://mathworld.wolfram.com/LambertW-Function....
mick's user avatar
  • 16.1k
4 votes
0 answers
80 views

Given a $3\times 3$ $\operatorname{adj} A$, find $A$

Given $\operatorname{adj}A=\begin{bmatrix} -1 & -2 & 1\\ 3 & 0 & -3 \\ 1 & -4 & 1 \end{bmatrix}$ . Find $A$. My Attempt We know that $|\operatorname{adj}A|=|A|^{n-1}\Rightarrow ...
Maverick's user avatar
  • 9,471
4 votes
0 answers
136 views

Transform multiplicative noise to additive noise with singular matrices

I have a stochastic differential equation with multiplicative noise $\alpha(t)$ \begin{equation} \dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X}...
J.Agusti's user avatar
  • 155
4 votes
0 answers
181 views

Are there applications for the inverse of the arc length of $ax^n$ and $a^x$? “Closed forms” found.

Based on: How to straighten a parabola? and Arc length of $x^n$ found using Hypergeometric function and series. Alternate representations and solution verification needed. Use: $$\text{ArcLength}(...
Тyma Gaidash ٠'s user avatar
4 votes
0 answers
135 views

Can I approximate the inverse by inverse of the approximation?

I haven't seen this done before, but I'm sure it's possible. Suppose $f$ is analytic on $[x_1,x_2]$ with range $[y_1,y_2],$ then its Taylor series converges on this interval and it is branch-...
StackQuest's user avatar
4 votes
0 answers
59 views

How many elements in the inverse of an $n\times n$ positive matrix can be positive?

Find all possible numbers that of positive elements in the inverse of an $n\times n$ positive matrix. For $n=2$, that is only $2$. This is because the inverse of a $2\times2$ positive matrix is of the ...
dodicta's user avatar
  • 1,423
4 votes
0 answers
287 views

Exact relation between kernel and invertibility of a differential operator

Let's assume we have a differential operator $D$ and a differential equation \begin{equation} Df=j \end{equation} What is the connection between invertibility, the kernel of the operator and the ...
NicAG's user avatar
  • 661
4 votes
0 answers
110 views

Inverse of perturbed Kac-Murdock-Szegö matrix

A Kac-Murdock-Szegö (KMS) matrix is a matrix of the form $A_{ij}=\rho^{|i-j|}$ for $i,j=1,2,\ldots,n$ and $\rho\neq1$. The inverse of $A^{-1}$ is well known, see e.g. https://journal.austms.org.au/ojs/...
user_goldeneye's user avatar
4 votes
0 answers
89 views

Number of Invertible Elements

How do you find the number of (and how to construct) invertible elements of the polynomial ring $F_2[X]/⟨X^n−1⟩$ for general $n$? I know the factorization of $X^n-1$ is useful but I'm not sure how to ...
Math_Finder 's user avatar
4 votes
0 answers
673 views

Woodbury Matrix Inversion

I am trying to invert a matrix using Woodbury identity. The inversion using Cholesky decomposition has the following pseudo-code: For $t=1,2,...$ $(1)\;\; \text{Read}\;x_t\in\mathbb{R}^n$ ...
Waqas's user avatar
  • 369
4 votes
0 answers
766 views

Inverse of matrix after updating diagonal?

Let $A$ be a real symmetric positive-definite matrix, with known inverse $A^{-1}$. Is there an efficient algorithm to compute $(A+R)^{-1}$, where $R$ is a real diagonal matrix? Assume that $A+R$ is ...
a06e's user avatar
  • 6,729
4 votes
0 answers
199 views

spin projector in inverted matrix

The following matrix $A$ is, \begin{equation} A= \begin{bmatrix} a+b-\sigma\cdot \textbf p & -x_1 \\ x_2 & a-b-\sigma\cdot \textbf p \end{bmatrix} \end{equation} The inversion of matrix $A$ is,...
Aschoolar's user avatar
  • 466
4 votes
0 answers
225 views

Pseudo-inverse with minimal number of non-zero entries

I'm looking for a way to compute the pseudo-inverse of a matrix (not the Moore-Penrose, but any other) with the minimal number of non-zero entries (maximum number of zero entries). In MATLAB, the the ...
Dorian's user avatar
  • 155
4 votes
0 answers
379 views

Need to improve upper bound for $\| (uv^T + B)^{-1} \|$ (Sherman-Morrison formula)

I have a matrix $A \in \mathbb{C}^{n \times n}$ in the form $A = uv^T + B$, where $u,v \in \mathbb{C}^n$ and $B \in \mathbb{C}^{n \times n}$. I know $A$ is invertible and want to find an upper bound ...
Integral's user avatar
  • 6,564
4 votes
0 answers
636 views

Inverse of block triangular matrix

How to find the pseudo-inverse of the following block lower triangular matrix? $$X=\begin{bmatrix} A & 0 \\ c & d \\ \end{bmatrix}$$ Where $A$ is a $n\times n$ lower triangular matrix, $d$ is ...
Astro's user avatar
  • 357
4 votes
0 answers
62 views

Proving whether the following are groups or not.

In each case, I am asked to decide whether the indicated pair is a group or not. If so, prove it; if not, show which group axiom fails. (a) $(\dfrac{1}{2}\mathbb{Z}, +)$ where $\dfrac{1}{2} \mathbb{Z}...
letsmakemuffinstogether's user avatar
4 votes
1 answer
468 views

Closed form for elements of inverse matrix of lower triangular matrix of any size

If we have a lower triangular matrix $$A=\left(\begin{array}{rrrrr}a_{1,1}&0&0&\cdots&0\\a_{1,2}&a_{2,2}&0&\cdots&0\\a_{1,3} &1_{2,3}&a_{3,3}&\cdots&0\\ ...
Avi's user avatar
  • 1,790
4 votes
0 answers
286 views

Eigenvectors of difference of inverse matrices

I have two matrices $A$ and $B$, symmetric and positive semi-definite (in fact, they are covariance matrices), and I am interested in computing the eigenvectors of the matrix $A^{-1}-B^{-1}$. From ...
gip's user avatar
  • 41
4 votes
0 answers
1k views

Inverse of identity plus scalar multiple of matrix

Given the matrix $M = ( I + \alpha D P )$, where $I$ is the nxn identity, $D$ is nxn symmetric and invertible, $P$ is nxn symmetric but not always invertible, and $\alpha$ is a scalar, is there a ...
Jobie's user avatar
  • 41
3 votes
0 answers
177 views

Is the conditional inverse a well defined inverse for matrices?

I'm a second year Computer science student and in my one mathematical statistics module my professor mentioned the idea of a Conditional inverse of a matrix and a Generalised inverse. He listed the ...
ZOfficial's user avatar
3 votes
0 answers
33 views

How many elements of the inverse of an $n\times n$ matrix of natural numbers can be the same as itself?

What is the maximum number of elements of the inverse of an $n\times n$ matrix consisting of natural numbers $(\geq1)$ that are identical to itself? This question arose from my previous question, How ...
dodicta's user avatar
  • 1,423
3 votes
0 answers
110 views

Is $0$ the Exponential Inverse?

For a while, I've been wondering why this pattern seems to allude to the fact that $0$ is one of the inverses of exponentiation. \begin{align} & x \cdot -1 = -x & \text{Inverse of addition} \\ ...
William Ryman's user avatar
3 votes
0 answers
469 views

Since moore penrose inverse matrix is an orthogonal projection matrix,what space does it project into?

We mark $A^+$ as a moore penrose inverse matrix $A^+A$,$I-A^+A$,$I-AA^+$,$AA^+$ These four matrices are easy to verify that they are symmetric idempotent matrices, that is, orthogonal projection ...
wit's user avatar
  • 109
3 votes
0 answers
103 views

Fastest and accurate way to inverse 2 and 3 variate Vandermonde matrix

I have a code which has a large for loop (100,000 iterations) and I need to inverse these matrices in every loop \begin{bmatrix} 1&x_1&x_1^2&y_1&y_1^2\\ 1&x_2&x_2^2&y_2&...
Duckduckcode's user avatar
3 votes
1 answer
50 views

Find the sum of inverses $\bmod p$ of numbers in range $[1,\frac{p-1}{2}]$

Let $p$ be an odd prime. Is it possible to compute $\sum_{k=1}^{\frac{p-1}{2}}k^{-1}\bmod p$ more efficiently than in $O(p)$?
abacabadabacaba's user avatar
3 votes
0 answers
104 views

Determinant and inverse of $M + M^T$ for a specific $M$.

Let $M$ be an upper triangular $n \times n$ matrix with entries only being 0 or 1 and having only ones on the diagonal. Let $N:=M + M^T$. Question: Can the determinant and the inverse (in case it ...
Mare's user avatar
  • 2,332
3 votes
0 answers
88 views

What statement can be made about $A^{-T}A$?

Given a matrix $A$ and the transpose of its inverse, $A^{-T}$, are there any useful properties of their product $A^{-T}A$? The only thing that came to my mind was $$\det(A^{-T}A) = \underbrace{\det(...
Tobias Kienzler's user avatar
3 votes
0 answers
151 views

Eigenvalues of the product of a subblock of a matrix and a subblock of the inverse

I have a block matrix (which is overall symmetric) $$ M = \begin{bmatrix} A & B \\ B^T & C \\ \end{bmatrix} $$ and it's inverse $$ M^{-1} = \begin{bmatrix} \tilde{A} & \...
Mete0r1t3's user avatar
3 votes
0 answers
159 views

Calculating the diagonal of $(I-Q)^{-1}$ efficiently

Motivation: I'm trying to write code to solve an equation efficiently. Directly calculating the result is easy, but involves matrix inversions that consume an impractical amount of memory at the scale ...
anjama's user avatar
  • 153
3 votes
0 answers
46 views

Probability that the sum of $k$ matrices is invertible

Suppose we have $k$ matrices over $\mathbb{Z}_q$ of size $n \times n$, where $q \gg k$ and the entries of the matrices are chosen uniformly at random. Assume $q$ to be prime and we do not pick any ...
chelsea's user avatar
  • 195
3 votes
0 answers
60 views

How to find the inverse of $A\otimes A + (I+D\otimes D)^{-1}(D\otimes D)$ without forming the Kronecker product?

Is there a good way to compute the inverse of Inverse of $A\otimes A + (I+D\otimes D)^{-1}(D\otimes D)$ that doesn't require forming the full Kronecker product? Here $A$ is symmetric, positive ...
wyer33's user avatar
  • 2,562
3 votes
0 answers
252 views

Solve $Ax=b$ for $A$

I have to solve the problem $b=(A-A^T)x$ for $A\in\mathbb R^{n\times n}$, where $b$ and $x$ are known. When we need to solve $Ax=b$, with $A$ not invertible, a known result is $x=A^\dagger b$ where $...
space_voyager's user avatar
3 votes
0 answers
1k views

SVD of matrix plus diagonal matrix and inversed

Assume we have a matrix $Z$ and we know its singular value decomposition. So, based on this, is there any trick for computing SVD of $(Z+cI)^{-1}$, where $cI$ is identity matrix, multiplied by some ...
Artem Moskalev's user avatar
3 votes
3 answers
89 views

Under what circumstances can we take this inverse matrix?

Assume we have a matrix: $$ (A^TB^{-1}A)^{-1}. $$ Under what circumstances can we simplify this to: $$ C^TBC? $$ And what would $C$ be? EDIT: Note that I am NOT assuming that $A$ is a square ...
user56834's user avatar
  • 13k
3 votes
0 answers
48 views

write the inverse of: $y= 3(4)^{2x+1} + 1$

write the inverse of: $y= 3(4)^{2x+1} + 1$ This is what I did: $ \begin{align} x=& 3(4)^{2y+1} + 1\\ x-1=& 3(4)^{2y+1} \\ \frac {(x-1)} 3=& (4)^{2y+1}\\ \log((x-1)/3)=& (2y+1)\log(...
Timmy's user avatar
  • 107
3 votes
2 answers
512 views

Inverse of the anti-commutator

Let $B \in M_{n \times n}(\mathbb{R})$ be a fixed $n \times n$ symmetric positive matrix. Consider the anti-commutator map $$f:M_{n \times n}(\mathbb{R})\to M_{n \times n}(\mathbb{R})$$ via $$f(C)=...
ZQ Wan's user avatar
  • 535

1
2 3 4 5
19