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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What properties do I have if I know $f$ and $f^{-1}$inverse are differentiable?

My goal is to show that $(f^{-1})'(y) = 1/[f'(f^{-1}(y)]$ for all $y$ in $(a,b)$. I have no idea where to start. I know that $f^{-1}$ and $f$ are differentiable.
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1answer
138 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 ...
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7answers
147 views

$f \circ g =\operatorname{ id}$ and $g \circ f \neq \operatorname{id}$?

Give two functions $f$ and $g$ s.t. $$f \circ g = \operatorname{id}$$ but $$g \circ f \neq \operatorname{id}$$ or a proof that this is impossible. This must be trivial, but I can't figure it out :) ...
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2answers
35 views

Showing that $\mathcal{G}(\ell_2)$ is not dense in $\mathcal{B}(\ell_2)$ via the right shift

This is my question: Is $\mathcal{G}(\ell_2)$ is dense in $\mathcal{B}(\ell_2)$? I am attempting to show that it is not by showing that the right-shift - call it $T:\ell_2 \rightarrow \ell_2$ - ...
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52 views

Inverse Relation of Irreflexive Property.

We are taking the inverse of relation to check that inverse of R is transitive, reflexive , symmetric and anti-symmetric to as it is on R (not inverse).. My question is that why we are not taking the ...
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0answers
48 views

Abscissa of absolute convergence of a Dirichlet series

I'd like some help to prove the following theorem : Let $\sum_{n \geq 1}\frac{f(n)}{n^s}$ and $\sum_{n \geq 1}\frac{g(n)}{n^s}$ be two Dirichlet series with respective abscissas of absolute ...
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57 views

Is this notation for inverse functions bad?

I'm trying to find useful notation for inverse functions that isn't too much in conflict with other notation already in use, but I'm wondering if this notation will come back and bite me in the ...
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21 views

Compute new inverse when old inverse and new and old matrix known

Say I have a matrix $M$ and know its inverse $M^{-1}$. Then every element changes so that $M'=M+(M'-M)$. Is there a fast way to find $M'^{-1}$ from this information? That is without computing the new ...
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34 views

The relation of domain and image of a function and its inverse

Theorem: Let both $f$ and $f^{-1}$ be functions. $\newcommand{\dom}{\operatorname{dom}}\newcommand{\im}{\operatorname{im}}$ Then $\dom(f) = \im(f^{-1})$ and $\dom(f^{-1}) = \im(f)$. Let $f: X ...
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1answer
18 views

Inverse of a function involving a Jacobian.

Why is it true that if the inverse of both $ \tilde{f} $ and $ f $ exists then: $$ \tilde{f}\left(\vec{x}\right) = [Df(x_{0})]^{-1} f(\vec{x}) $$ $$ \implies \tilde{f}^{-1}(\vec{x}) = ...
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86 views

The inverse of $(I-A)$ and the spectral radius of a nonnegative $A$ matrix

Suppost that $A$ is a nonnegative matrix, and let denote the identitiy matrix with $I$ and the spectral radius of $A$ with $\rho(A)$. Note that because $A$ is nonnegative according to the ...
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1answer
28 views

How to verify an algebraic structure is a ring

I have a problem which ask me to verify that to structures are rings. However, I'm unsure of how exactly to check each property. I believe that the first is closed but not sure how to check the ...
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1answer
22 views

Linear maps, inverses and associated matrices?

This is likely a very simple question but if we have a linear map $f$ with an associated matrix $A$ is it a necessary and sufficient condition that for $f$ to have an inverse then $A$ must also have ...
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4answers
87 views

Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$

I have a function: $$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$ With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...
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1answer
44 views

Inverse matrix as a sum of matrix powers [duplicate]

I have matrix $ A\in \mathbb{C}^{n x n}$ and $A$ is invertible. How can I show that coefficients $c_0,...,c_{n-1}$ exist : $A^{-1} = c_0I+c_1A+...+c_{n-1}A^{n-1}$ I tried to solve it first by ...
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1answer
62 views

What is the Moore-Penrose pseudoinverse for a hermitian block-matrix with one zero block?

Given a block matrix of the form \begin{pmatrix} A & B^* \\ B & 0 \end{pmatrix} where $A$ is singular (otherwise one could simply use the well-known block matrix inverse), is there a ...
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1answer
35 views

Why are those equivalent transformations of inverse functions not the same thing?

Why are $\frac{1}{f}=\frac{1}{g}+\frac{1}{b}$ and $f=g+b$ not the same thing?
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1answer
28 views

Polynomial has right inverse implies invertible?

If $p:\Bbb R\rightarrow \Bbb R$ is a real polynomial such that $p$ has a right inverse $q$, does it follow that $p$ is invertible? That is, must $q$ also be a left inverse of $p$? The question ...
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1answer
31 views

Two roots of $\arcsin(x)$ in the range $[0,2 \pi]$

I am baffled with how to write the two roots of arcSin$(x)$ in the range $[0,2 \pi]$, while $x \in [-1,1]$, such that one root can be directly calculated in terms of the other root. For instance, we ...
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48 views

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$.

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$. Proof: Let $f(x) = \sin x$, for $x$ in $(-1,1)$. Then let $x_{0}$ be in (-1,1). Then $f'(x_{0})$ = ...
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5answers
65 views

True or false? Prove it.

If $A$ is an $n\times n$ invertible matrix and $B$ is an $n\times m$ matrix, then $\operatorname{rank}(AB) = \operatorname{rank}(B)$. Is this true or false? I've tried proven that if $B=0$, then ...
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1answer
36 views

Fourier Transform Inverse of 1 / (jw - a)

I want to find the inverse fourier transform of $$ \frac 1 {j \omega - 1} $$ The fourier transform of $$ e^{-at} u(t) $$ is $$ \frac {1}{j \omega + a} $$ This result if true ONLY if a > 0. If a ...
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212 views

Inverse Square law problem, how to calculate for distances.

i've got a bit of a problem with the inverse square law (I1/I2=D2 squared/D1 squared)(Where I=intensity and D=distance) I need to change a distance from 1000mm to 400mm (I'm a Radiographer). Most of ...
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28 views

Inverse of a particular operator

I need help finding the inverse of the following operator. I am not sure about how to start. Any help would be hugely appreciated. Operator: $( I + \frac{\partial^2}{\partial x^2})$ Edit: I ...
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1answer
64 views

Finding derivative of the inverse without the inverse

We are given a function $$f(x)=4\arcsin(\sqrt{x})+2\arcsin(\sqrt{1-x})$$ The derivative of $f$ is: $$f'(x)=\frac{1}{\sqrt{x-x^2}}$$ I would like to find the maximum value of $f^{-1}$. I think I have a ...
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1answer
40 views

quadratic equation modulo some number

I read a post that $$ax^2+bx+c \equiv 1 \pmod p$$ can be solved in a similar way we solve a simple quadratic equation, just by replacing division by $2a$ by modulo inverse of $2a$ and square root of ...
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56 views

Catch 22 situation involving inverting a function and finding the range of the function.

Let $f(x) = \sqrt{x+5} - \sqrt{x-5}$ Calculating the inverse: $y = \sqrt{x+5} - \sqrt{x-5}$ $y + \sqrt{x-5} = \sqrt{x+5}$ $y^2 + x - 5 + 2y\sqrt{x-5} = x + 5$ $\frac{(10 - y^2)^2}{4y^2} + 5 = x$ ...
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90 views

Prove that $f^{-1}$ exists and is differentiable on $(0, ∞)$ for $f(x) = x^2e^{x^2}$.

Let $f(x) = x^2e^{x^2}$, and assume that $(e^x)' = e^x$ for all $x$ in $R$. a) Prove that $f^{-1}$ exists and is differentiable on $(0, ∞)$. Proof: Suppose that $f(x) = x^2e^{x^2}$, then finding ...
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23 views

right inverse and supplement of kernel in a banach

For $T \in L(E,F)$ continuous surjective linear operator between Banach spaces $E$ and $F$ we have that : $Ker(T) $ admits a closed complement $L$ in $E \implies T$ admits a continuous right ...
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1answer
18 views

Question Regarding Inverses In a Function

Here is my current issue. Our teacher asked a question related to the finding of an inverse of 2. Here is all of the given information: Define "a cross b" as such: a ☢ b = ab + (a + b). Use this ...
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1answer
26 views

Inversion of a symmetric and positive definite matrix with or without a column and row

Suppose to have a symmetric and positive definite matrix $\boldsymbol{\Sigma}$ and suppose to know its inverse $\boldsymbol{\Sigma}^{-1}$. Let $\boldsymbol{\Sigma}_{+}= \left( \begin{array}{cc} ...
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37 views

Inverse Trig Functions, finding Domain and Range

I understand the restricted domains of inverse trig functions, but what about: I don't quite understand how to find the domain and range of this function.
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39 views

Inverse of the matrix product $\boldsymbol{A} \cdot\boldsymbol{S} \cdot \boldsymbol{A}^{T}$

If I have an $n\times n$ symmetric matrix $\boldsymbol{S}$ and a $m\times n$ matrix $\boldsymbol{A}$ is there any relation between $(\boldsymbol{A} \cdot\boldsymbol{S} \cdot \boldsymbol{A}^{T})^{-1}$ ...
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46 views

Solution for set of matrix equations involving an inverse

I am encountering the following set of three matrix equations for which I search a solution in terms of ${\bf M}\in\mathbb{R}^{N\times N}$ and ${\bf D}\in\mathbb{R}^{Q\times N}$, $${\bf M}{\bf W} = ...
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1answer
44 views

Find Invertible and NonInvertible Matrix

Can someone help me to understand this problem? I don't know where to begin. Find an invertible matrix $A$ and a noninvertible matrix $B$ both of which satisfy $$M^2=3M$$ Thanks, Rusty
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1answer
327 views

Find all right inverses of matrix A.

I'm given the matrix A where it's first row is $(2, -1, 3)$ and second row is $(1, 2, 1)$ and I'm told to find all the right inverses of it. First I tried doing A times a 3x2 vector B (just a vector ...
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212 views

How adjacency matrix shows that the graph have no cycles?

Let $G$ a directed graph and $A$ the corresponding adjacency matrix. Let denote the identity matrix with $I$. I've read in a wikipedia article, that the following statement is true. Question. Is it ...
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87 views

Fields over which a matrix is not invertible

I am trying to find the fields over which the matrix: $\left(\begin{matrix} 1 & 2 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & -2 \end{matrix}\right) $ is not invertible. I have ...
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1answer
107 views

Simple formula for a sieries like 1, 2, 5, 10, 20, 50, 100, …

I'm looking for a simple formula that will give a series that looks like this: $1; 2; 5; 10; 20; 50; 100; ...$ That means a function that will give this output: $f(1) = 1$ $f(2) = 2$ $f(3) = 5$ $f(4) ...
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84 views

Solve Inverse Linear Congruence

I want to solve Linear congrunece : 9x+2 ≡ 6(mod 1453) using inverse of 9 mod 1453. Inverse of 9 mod 1453 is 323. Now to solve it I subtract 2 from left and right side which gives me 9x ≡ 4(mod 1453), ...
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Bijection, and finding the inverse function

I am new to discrete mathematics, and this was one of the question that the prof gave out. I am bit lost in this, since I never encountered discrete mathematics before. What do I need to do to prove ...
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1answer
37 views

Psuedo-inverse of block low-rank, symmetric matrix?

I have a matrix that looks like $$ D = \left[ \begin{matrix} c_1aa^T & c_2ab^T \\ c_2ba^T & c_3bb^T \end{matrix} \right] $$ where $c_1, c_2, c_3$ are scalars and $a, b$ ...
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1answer
52 views

Finding inverse of $g(x) = \dfrac{3x + 1}{2x + g(x)}$

Find $g^{-1}(3)$ given $g(x) = \dfrac{3x + 1}{2x + g(x)}$ My Approach: \begin{align*} y & = \frac{3x + 1}{2x + y} && \text{(does $g(x)$ become $y$ also?)}\\ x & = \frac{3y + 1}{2y ...
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1answer
49 views

What are the possible values of $x$?

For what values of $x$ does this equation holds? $$2\arctan(x)=\arctan\left(\frac{2x}{1-x^2}\right)$$ The answer is $-1<x<1$ Why? How can we say this?
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40 views

Inverse of a triangular matrix of special form

How should I begin when I want to get an inverse matrix from this one? Progress I have tried to do it explicitly for small $n$. But now I am not able to deduce the general pattern with $n$ from ...
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1answer
327 views

Can A be singular? [duplicate]

Let $A\in \mathbb{C}^{n\times n}$ satisfy $$A^{2}+A+I=0 $$ Can A be singular? So I have: $$ (A-I)(A^{2}+A+I)=0\\ A^{3} = I \\ (\det A^{3}) = \det(I) \\ (\det A)^{3} = 1\\ \det A\neq 0 $$ So $A$ is ...
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1answer
38 views

If an analytic function $f: \mathbb{R}^2 \to \mathbb{R}^2$ is locally invertible at $(x_0, y_0)$, then $Df(x_0,y_0) \not = 0$.

I am trying to show that if an analytic function $f: \mathbb{R}^2 \to \mathbb{R}^2$ (i.e. $f$ satisfies the Cauchy-Riemann equations) is locally invertible at $(x_0, y_0)$, then $Df(x_0,y_0) \not = ...
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95 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 ...
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1answer
20 views

Find a right inverse of a map with gauss brackets.

I am having a composition of two maps: $$ f:\mathbb{R}->\mathbb{R_0^+},f(x)=x^2 $$ $$ g:\mathbb{R_0^+}->\mathbb{\mathbb{N}},g(x)=\lfloor x\rfloor $$ $$h=g\circ f:\mathbb{R}->\mathbb{N_0}$$ ...
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0answers
26 views

inverse hyperbolic function of a complex argument

It is not too difficult to prove that $f(z)=\cosh z$ is a bijection from $$\def\C{{\Bbb C}}D=\{\,z=x+iy\in\C\mid 0<y<\pi\,\}$$ to $$R=\{\,w=u+iv\in\C\mid v\ne0\,\}\cup\{\,w\in\C\mid ...