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.

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6
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153 views

How to invert this expression involving $\tanh^{-1}$?

I've got the expression: $ x = \tanh^{-1}(p) - \sqrt{\frac{2}{3}} \tanh^{-1}\left( \sqrt{\frac{2}{3}} p\right) $ How can I invert this function so I have a function $p(x)$? I thought about using ...
6
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0answers
71 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}...
5
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126 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 ...
5
votes
0answers
93 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^\...
4
votes
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43 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}...
4
votes
0answers
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 ...
3
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0answers
50 views

Time complexity of inverting an $n \times n$ matrix which is the sum of a rank-$m$ matrix and a full-rank diagonal matrix

I want to know the time complexity of inverting $K$, where $K$ is an positive-definite $n\times n$ matrix: $$K=\Lambda+Q$$, where $\Lambda$ and $Q$ are both $n\times n$ matrix, $\Lambda$ is a full-...
3
votes
0answers
30 views

On generalised inverse

Let $A$ be a positive matrix, may not be invertible. I define its generalised inverse as \begin{equation} A^- = \lim_{n\rightarrow \infty} \left( \frac{1}{n} I + A\right)^{-1}. \end{equation} Lets ...
3
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57 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 ...
3
votes
0answers
71 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 ...
3
votes
0answers
83 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 ...
3
votes
0answers
312 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. ...
3
votes
0answers
843 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}\| & = \sup\left\{\frac{\|A^{-1}b\|}{\|b\|...
3
votes
0answers
84 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. ...
3
votes
0answers
458 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
votes
0answers
51 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 ...
3
votes
0answers
117 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 ...
3
votes
0answers
50 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 ...
3
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0answers
1k views

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 ...
3
votes
0answers
141 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 ...
2
votes
0answers
25 views

Perturbations to a matrix causing drastic changes to matrix inverse.

I'm reading this article about matrix norms because I want to understanding the math behind SVD. One of the interesting issues it brings up quite soon is the effect of perturbations to a matrix on ...
2
votes
0answers
56 views

Derivative of $(\lambda I - A)^{-1}$ with respect to $\lambda$

Is need to work with $\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u)$. Is it true that: $$\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u) = -v^{T}\frac{d}{d\lambda}(\lambda I - A)^{-1}u$$ ...
2
votes
0answers
29 views

Is this geometric Interpretation of $Q^T$ being orthonormal if $Q$ is orthonormal valid?

I was reading the book - Linear Algebra and its Applications, when I saw - Remark 2. Since $Q^T = Q^{-1}$, we also have $QQ^T = I$. When Q comes before $Q^T$, multiplication takes the inner ...
2
votes
0answers
45 views

'Stable' Ways To Invert A Matrix

So lets say that I need to invert a matrix that is generally dense and is poorly conditioned. What are some ways I can get an accurate inverse? Here are my candidates: SVD Inverse Inverse Via ...
2
votes
0answers
48 views

Finding the multiplicative inverses in fields

Let's say I have the field $F_{11}$. Why does $2$ have the multiplicative inverse $6$? In some of the examples I have, let's say we are looking $F_5$, why are values up to only $2$ considered? So ...
2
votes
0answers
40 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: $$M_\text{right}^{-1} = M^T(MM^T)^{-1}$$ ...
2
votes
0answers
44 views

Matrix inverse series expansion

I want to prove that when $I+K$ is invertible, $$(I+K)^{-1}=I-K+o(K)$$ to establish that the matrix inverse function has derivative $-I$ at $I$. My hope is that this identity carries over from $\...
2
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0answers
81 views

Inverse of generalized arrow matrix $A = M^T * M + I$

If we have the following linear system: Ax=b And matrix A is created by multiplying a rectangular matrix with it's transpose: $A = M^T * M + I$ What is the best method to solve for x for different b ...
2
votes
0answers
86 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 ...
2
votes
0answers
33 views

Continuity at an inner point of an interval implies the continuity of the inverse

If a function $f$ with domain interval is $1-1$ and continuous at an inner point a of the interval its inverse is continuous at $f(a)$?
2
votes
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26 views

Inverse Fourier transform using laplace

We have to solve the inverse FT of $$\frac{1}{1+4w^2}$$ I tried to do the synthesis but got mediocre results. However this term screams laplace to me. I can see a sine in there. The last lecture they ...
2
votes
0answers
45 views

How can I use the Lagrange Inversion Theorem?

I have a function $f(x)= x(\ln(x\ln x))$ and I want to use the Lagrange Inversion theorem to find its inverse $g(x)$ centered around a point $a$. The formula states that: $$g(x)=a+\sum_{n=1}^{\infty} \...
2
votes
0answers
36 views

Differential Equation for Inverse of a Function given the ODE for the function

Assume we have and ODE for a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ in the form of \begin{equation} \mathbf{J}f \times F(x)=D(\lambda_i)\times f(x) \end{equation} where $\mathbf{J}$ denotes ...
2
votes
0answers
52 views

another number group?

I noticed that for each basic increasing binary function (addition, multiplication, and exponentiation) its inverse (or just a inverse) of certain values adds more number types to the number line (or ...
2
votes
0answers
30 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 ...
2
votes
0answers
144 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\\ &...
2
votes
0answers
59 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 + \operatorname{trunc}\...
2
votes
0answers
114 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\\ ...
2
votes
0answers
417 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 $p=...
2
votes
0answers
197 views

What is the inverse kernel to this integral transform

What is the associated inverse kernel to the integral transform $T$ defined by \begin{align*} (Tf)(u) & = \int_{-\infty}^{0} \hat{f}(s)\exp((2i\pi+c)us)\ ds + \int_{0}^{+\infty} \hat{f}(s)\exp((2i\...
2
votes
0answers
44 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 ...
2
votes
0answers
779 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 ...
2
votes
0answers
464 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 ...
2
votes
0answers
243 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 \infty}\mathbf{...
2
votes
0answers
137 views

inverse of Vandermonde's Matrix without using determinants

I want to show, that the Vandermonde's Matrix $A:==\begin{pmatrix}1&x_0&\cdots&x_0^d\\1&x_1&\cdots&x_1^d\\\vdots&\vdots&\ddots&\vdots\\1&x_d&\cdots&x_d^...
2
votes
0answers
142 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 ...
2
votes
0answers
51 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 $$c(\tilde{b})=1-\frac{1}{(1-\tilde{b})\exp\left(\frac{2\tilde{b}}{(1-\tilde{b})}\right)}$...
2
votes
0answers
29 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 ...
2
votes
0answers
66 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
votes
0answers
150 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 ...