Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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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 ...
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224 views

Inversion of elliptic integral

I have an equation of the type $$ p=\int_0^b\sqrt{\left(a^2-x^2\right)\left(b^2-x^2\right)}dx, $$ in which $a$ and $b$ (with $a>b>0$) are (known) functions of some parameter $H$ (such that it is ...
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79 views

What function satisfy: $f(x)+f^{-1}(x)=2x$?

What function satisfy: $f(x)+f^{-1}(x)=2x$? I have tried to substitute $x=f(x)$ to get $f^{(2)}(x)+1=2f(x)$ and subsequently plug in values to try to find $f(x)$ but to no avail. Please help thank ...
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50 views

Is there always a smooth variant of a homoeomorphism 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 ...
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78 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 ...
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83 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 ...
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50 views

The inverse of x!

what is the inverse of a factorial function? Its is not continuous but is modeled by the gamma function which is continuous so must have a inverse any research leads to the inverse gamma function that ...
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61 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
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72 views

Inverse of $x^2+\log^2\cos x$

I'm looking for the inverse of $$f(x)=x^2+(\log\cos x)^2$$ Where $f$ is defined from $[0,\pi/2)$ It dosen't have to be closed form, a sum, an integral or some special functions would be of interest ...
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89 views

The inverse of a transpose matrix to “cancel” the transpose?

When it comes to solving and equation containing matrices I don't always understand some of the rules involved. In particular, I am trying to figure out the derivation of the Gauss-Newton algorithm. ...
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81 views

Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
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316 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 ...
3
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44 views

Error bounds in representing a vector using a truncated Moore-Penrose biorthogonal basis

I was reading and trying to reproduce the results in the arXiv preprint of Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression by Asaf ...
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93 views

Ultrametric matrices and their inverse

A non-negative square matrix $A$ is ultrametric iff: $A(i,i)>\{A(i,k),A(k,i)\}\forall k,i$ $A(i,j)\geq\min\{A(i,k),A(k,j)\}\forall i,j,k$ It is well-known that the inverse of non-negative ...
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1k views

Left inverse iff injective; right inverse iff surjective

For a function $f:A\to B$, the function $g:B\to A$ is called: a left inverse for $f$ if $g\circ f$ is the identity on $A$ (i.e., $g\circ f = {\rm id}_A$); and a right inverse for $f$ if ...
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49 views

Finding the number of the real roots of $a^x=g(x)$ where $g(x)$ is the inverse function of $f(x)=a^x$

Question : Let $a$ be a constant which satisfies $0\lt a\lt 1$. Letting $g(x)$ be the inverse function of $f(x)=a^x$, then find the number $N$ of the real roots of $f(x)=g(x)$. Motivation : This is ...
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130 views

Sparse matrix inverse multiplied by sparse matrices

I have the equation $\bf E = Y D^{-1} Y^\top$. $\bf D$ is a potentially large sparse $m \times m$ matrix, and $\bf Y$ is a sparse $n \times m$ matrix, where $n \ll m$. Is there a particularly ...
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20 views

Closest line to point after non-linear map

I have a map on a vector space $M(\vec{r})$, defined as below. All components (vectors, matrices, everything) are reals in the unit range $[0,1]$. The map $M(\vec r)$ is defined as the sum of an ...
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40 views

Inverse of sum of 3 matrices

I need a way to compute the inverse of the sum of three matrices: $(A + BB^T + \beta I)^{-1} $ where $I$ is identity and $\beta$ is a constant. I am not very familiar with linear algebra, but a ...
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45 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\\ ...
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48 views

Inversion of a pairing function

I was reading this question on this site and I saw that the following pairing function was mentioned (a modified version of Cantor function): $$\langle x, y\rangle = x * y + ...
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86 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_{2,1}&a_{2,2}&0&\cdots&0\\a_{3,1} &1_{3,2}&a_{3,3}&\cdots&0\\ ...
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152 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
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171 views

Fastest way to find modular multiplicative inverse

I am looking for a fast way to find the modular multiplicate inverse of an integer $a$ in mod $p$. I am mainly interested in ...
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42 views

Determining $f^{-1}(3)$ without knowing $f^{-1}(x)$ but given $f(1)=3$ and $f'(x)>0$.

I have a continuous function $f(x)$ and I want to find $f^{-1}(3)$, but I can't find $f^{-1}$ directly. I know that $f(1)=3$ and $f'(x)>0$ for all x. Because the function is continuous and always ...
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384 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
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95 views

Self-inverse matrices with integers with pairwise different absolut values.

Let A be a self-inverse matrix ($A=A^{-1}$) with integer values such that no two integers have the same absolut value. Let M be the maximum of the absolut values (maximum-norm) of A. Which M is the ...
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93 views

Is there an efficient method to find all the self-inverse matrices with integers in a given range?

Given $n$ and a range, for example $[-10,10]$, is there an efficient method to find all $n \times n$-matrices $A$ with integers in the given range, which are self-inverse (that means the equation ...
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117 views

matrix inverse and limit

I would like to get a better understanding of limits and matrix inverses, specifically the relationship between: $\lim_{k\rightarrow \infty}(\mathbf{A}^{-1})$ and $(\lim_{k\rightarrow ...
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426 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
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100 views

Pseudo-inverse of an underdetermined Toeplitz matrix

I have an undetermined Toeplitz matrix (more columns than rows). For example: \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 ...
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47 views

Integrating inverse functions

I'm trying to integrate the following: $$\int_0^1 \left[\frac{c}{(1+c^{-1}(\tilde{b}))}\right]dc$$ If it helps ...
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27 views

probability subspaces that make entropy function equal to a constant value

Given the entropy fucntion: $$ H = -\sum_i^n p(i) \ln(p(i))\,.$$ where $p(i)$ are probabilities and $n=4$, I would like to know all the points in the probability space that make $H = k$, being $k$ a ...
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61 views

Taking the (pseudo)inverse of a monoid operation.

Let $M$ be a monoid with binary operation $f : M \times M \to M$. I'm interested in functions $g : M \to M\times M$ that obey the property: $$ f(g(m)) = m $$ I want to understand what all of the ...
2
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104 views

Continuous function that is invertible in one argument---is its inverse continuous in both arguments?

Suppose that $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous function and that it is invertible in its second argument, i.e. for every $x \in \mathbb{R}$, $f(x,\cdot)$ is invertible with ...
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138 views

Question about the Miller Theorem on inverse of sum of two matrices.

The following is a well known theorem on the inverse of $(A+B)$. (Link to the paper: http://www.jstor.org/stable/2690437) Theorem. Let $A$ and $A+B$ be nonsingular matrices, and let $B$ have rank ...
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51 views

Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
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139 views

Modular Inverse over some given finite field. Which method is more efficient?

I'm trying to do division in some given finite field (let's say mod p). I have 2 Python methods here that are currently doing that, but I'm not sure which is better or if 1 or both is simply wrong. ...
2
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63 views

About the functional inverse of integrals and infinite products.

It seems $\cos(x)$ and $\sin(x)$ are the only entire functions, that are the functional inverse of an integral of some elementary function $f(x)$ , such that they have a simple infinite product ...
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405 views

Proving invertibility of matrices using banachs lemma

I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...
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160 views

Optimization problem about large matrices

I'd like to solve the following optimization problem: Find non-negative scalar $a$, $b$, $c$ to minimize $\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$ where ...
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710 views

Inverse Jacobian matrix of spherical coordinates

I found inverse transformation from spherical coordinates to cartesian coordinates (on $x>0$, $y>0$ and $z>0$). I have $$ r = w_1(x,y,z) = \sqrt{x^2+y^2+z^2} $$ $$ \theta = w_2(x,y,z) = ...
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How to calculate the submatrix inverse with prior knowledge of matrix inverse?

Given $A\in \mathbb{N}^{n\times n}$, then $A(\mathcal{I})$ is defined by first deleting the those columns with index in $\mathcal{I}$ and then extracting the first $n-|\mathcal{I}|$ rows. Note that ...
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379 views

Multivariable Inverse Function problem

Consider the system of equations $$\left\{\begin{align*} &x^5 v^2 + 2y^3 u = 3\\ &3yu - xu v^3 = 2\;. \end{align*}\right.$$ Show that near the point $(x,y,u,v) = (1,1,1,1)$, this system ...
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435 views

Finding inverse of function without knowing function?

This question has a programming application, but I thought it would be more appropriate (and educational) to ask here first and get some basic understanding. So my use-case will refer to some ...
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46 views

Inverse Laplace Transform using Hetnarski's Algorithm

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
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24 views

Ideas for expressing the inverse of matrix quadratic form $CAC^T$

I want to find an expression for the inverse of the matrix system $Z=CAC^T$, where $A \in \mathbb{C}^{n \times n}$ is block diagonal with dense blocks, and $C \in \{-1,0,1\}$ with dimension $m \times ...
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16 views

Inverse of a Bijective Bivariate quadratic function or polynomial

I am looking for some general way to invert a bijective quadratic polynomial of the form $$ f(x,y)=A_0x+A_1x^2+Axy+B_0y+B_1y^2+Byx $$ where the coefficients may or may not be in the same ring as the ...
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36 views

Inverse of $R^T D R$ where $R$: rectangular and $D$: diagonal

Is there any formula for the following triple product: $$(R^T D R)^{-1}$$ where $R$ is rectangular and $D$ is diagonal? The real situation is like this. I have the equation $$Ax = b$$ which is a ...
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55 views

Is there a closed form expression for $(A^T\Sigma A)^{-1}$ when $A$ is not square?

I need to find the inverse of the matrix $A^T\Sigma A$. Matrix $A$ has dimensions $5\times 2$. Matrix $\Sigma$ has dimensions $5\times 5$, and it is symmetric and positive-definite. I need to ...