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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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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15 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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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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54 views

Formula for Area of parallelogram induced by linear operator

I'm given that the linear operator $L: \mathbb R^2\to\mathbb R^2$ is invertible. The set (u,v) is a linearly independent set in $\mathbb R^2$. I must find a formula for the area of the parallelogram ...
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20 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
13 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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56 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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12 views

Using the inverse of the matrix find all the solutions of the following systems of equations?

I found the inverse using row operations and the identity matrix but I dont know where to go from here. Can someone direct me please ?
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23 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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13 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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34 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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24 views

Proofs with algebraic structures (rings)

If one is given a ring $R$ with a unity $u$, what are the steps one would have to take to prove that some element of $R$ named $s$ has a multiplicative inverse, where $-s$ also has a multiplicative ...
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18 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
21 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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18 views

Compute the inverse fourier transform of $ e^{-Af} $ and $ e^{-A\sqrt{f}} $

I want to compute the inverse fourier transform of $ e^{-Af} $ and $ e^{-A\sqrt{f}} $, where $A$ is a constant and $f$ is frequency. In the case of $ e^{-Af} $, I tried to solve it from the Fourier ...
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35 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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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
16 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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26 views

Differential operator and its inverse

suppose we have a differential operator $D$ in terms of a variable $x$ and its inverse is denoted by $D^{-1}$ , then is it possible that $DD^{-1}=\delta (x)$ or $D^{-1}D=\delta (x)$? If so, then what ...
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24 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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24 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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44 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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19 views

Fast inversion of triangular matrix

I wrote the code below to invert an upper triangular matrix, avoiding any possible multiplication/subtraction by zero. It just uses $\frac{1}{6}n^3+\ldots$ flops instead of $n^3+\ldots$ flops. ...
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21 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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47 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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41 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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11 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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16 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
19 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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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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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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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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35 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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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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76 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. Statement. $I-A$ ...
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60 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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82 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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49 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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19 views

Calculation of Moore Penrose Inverse

I wanted to know if there is any way to find Moore Penrose inverse of a matrix by row transformation or Gauss-Jordan method. As, I wanted to work it out on a paper. Other methods like SVD requires ...
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78 views

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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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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50 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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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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322 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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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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68 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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17 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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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 ...
3
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1answer
29 views

When is Block-Partitioned Matrix Invertible?

Suppose I have a block partitioned matrix \begin{equation} \begin{bmatrix} \mathbf{X}_1^{\top}\mathbf{X}_1 & \mathbf{X}_1^{\top}\mathbf{X}_2 \\ \mathbf{X}_2^{\top}\mathbf{X}_1 & ...
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21 views

What is a quickest way to find inverses of functions of two variables?

Suppose I have $X_1 = aY_1+bY_2$ $X_2 = cY_1+dY_2$ How is the quickest (or most efficient way ) to find the inverse functions? The current way I am doing it is attempting to solve for Y1 in the ...