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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40 views

How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$

Question: Let $E=\{(x,y): 0<y<x \}$ set $f(x,y)=(x+y, xy)$ for $(x,y)\in E$ a) How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$ And how to find formula for ...
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1answer
313 views

Cholesky/LU decomposition from matrix and its inverse?

Usually, we have a matrix $A$ and want to calculate the $LU$ (or sometimes Cholesky, depending on $A$'s properties) decomposition. This is often the hard part. Now, if we have the $LU$ decomposition ...
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2answers
91 views

Inverting a special matrix

Consider matrices $A$ and $B$ of the forms below: $$A = \lambda \cdot I$$ $$B = \beta \cdot \pmatrix{ 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & ...
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1answer
122 views

Inverse function of $y=x+kx^3$

I want to invert the following function with respect to $x$: $$f(x, k)=x+k x^3$$ where typical values for $x$ are between $0$ and $100$ and typical values for $k$ are between $-0.00005$ and ...
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1answer
216 views

Inverse of $(A + B)$ and $(A + BCD)$?

Consider $A$ as an arbitrary matrix and $B$ as a symmetric matrix. Since $B$ is symmetric, therefore, it can be written as a $\Gamma \Delta \Gamma'$, where $\Delta$ is a diagonal matrix with ...
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2answers
98 views

Inverse of an $n\times n$ matrix

I'm curious to know the matrix form of the inverse of an $n\times n$ matrix. Also, how many operations will be needed to compute it?
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1answer
49 views

Closed forms for $\lim_{x\rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$

Im looking for closed forms for $\lim_{x \rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$ where $x$ is a positive real, $a$ is a given real, $p$ is the set of primes such that the ...
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1answer
204 views

Invertibility of a linear operator on a Hilbert space.

Let $H$ be an infinite dimensional Hilbert space over $\mathbb C$, $T$ be a continuous linear operator of $H$, $r(T)=\sup_{||x||=1}|(Tx|x)|$ be the numerical radius of $T$, and $z\in \mathbb C$, such ...
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0answers
55 views

Looking for examples where $f(z)=\operatorname{inv} \int_{0}^{z} g(z)\, dz$ with $f(z)$ entire and $g(z)$ not meromorphic.

I'm looking for examples where $f(z)=\operatorname{inv}\int_{0}^{z} g(z) \, dz$ with $f(z)$ entire and $g(z)$ not meromorphic. For clarity, by $\operatorname{inv}$, I mean the functional inverse. ...
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1answer
53 views

Need an easy CDF for Inverse transform sampling

I want to use inverse transform sampling to generate some random numbers, which all fall into a given interval $(0,x_{max})$. The numbers are not necessarily distributed evenly but can be "skewed". I ...
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1answer
68 views

Distribution of the Inverse of a Random Variable

I am trying to figure out how to find the distribution of the inverse of a random variable. Say, $Y=X^{-1}$ where X can take negative values. The two ways I know to find the distribution of a random ...
3
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2answers
308 views

Right Inverse for Surjective Function

Prove that if $f:X\to Y$ is a surjective function between sets, then there must exist a function $g:Y\rightarrow X$ such that $f\circ g=1_Y$. I know that the identity function is onto, and if $f$ ...
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1answer
396 views

Finding the inverse of a matrix using elementary matricies

Can somebody help me understand what exactly is being asked here? I understand how to construct elementary matrices from these row operations, but I'm unsure what the end goal is. Am I to assume that ...
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2answers
94 views

how to prove this equality

There are two equalities, $\sinh({\cosh }^{-1}x)=\sqrt { {x }^2-1 }\quad (x>1)$ $\cosh({\sinh}^{-1}y)=\sqrt{1+{y}^2}$ prove this equality please.. how to prove it? i cannot try it.. also, ...
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1answer
146 views

How to prove that a circle passing through the center of the circle of inversion invert to a line?

link to the referenced picture: http://www.flickr.com/photos/90803347@N03/9220374271/ In order to prove the Arbelos Theorem, as in the picture above, one need to prove that the semicircle $C$ invert ...
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0answers
50 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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2answers
82 views

Moore-Penrose Inverse and Standard Inverse

I have read that the Moore-Penrose inverse $A^+$ of a matrix $A$ is the same as the standard inverse $A^{-1}$ in the case $A$ is a square, invertible matrix. Is there any relation between the ...
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2answers
53 views

Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?

Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how? This seems like an easy question, but ...
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3answers
266 views

Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
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1answer
245 views

Matrix similar to its inverse

I have this problem: $A$ is an $n \times n$-matrix, its characteristic polynomial is $P(X)=(X-1)^n$. Prove that $A$ is similar to its inverse. How do you solve it? I really don't know.
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1answer
193 views

How to calculate the inverse of a point with respect to a circle?

The theory said: The inverse of a point $P$, with respect to a circle centered at $O$ and has a radius $r$, is the point $P'$ such that The three points $O$, $P$ and $P'$ are colinear. $OP \times ...
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3answers
324 views

Is there an inverse to Stirling's approximation?

The factorial function cannot have an inverse, $0!$ and $1!$ having the same value. However, Stirling's approximation of the factorial $x! \sim x^xe^{-x}\sqrt{2\pi x}$ does not have this problem, and ...
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2answers
84 views

1) Suppose $g \circ f$ is one-to-one. Prove that $f$ is one to one. Does $g$ have to be also one-to-one?

Let $f: A \to B $ and $g:B \to A.$ 1) Suppose $g \circ f$ is one-to-one. Prove that $f$ is one to one. Does $g$ have to be also one-to-one? 2) Suppose $g \circ f$ is onto. Prove that $g$ is ...
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2answers
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Singularity in matrix when inverting in Matlab

As data I get a matrix A but in my algorithm I need to work on its inverse. What I do is: C = inv(A) + B; Then in another line I update A. In the next cycles I ...
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1answer
212 views

Bounded operator inverse, norm and spectrum

I need help with an operator, I am not very good at functional analysis and need to find some properties of following operator: $X=C[(0,1)], A \in B(X); A[f(t)]=f(t^2)$ 1. I need to show that an ...
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3answers
398 views

How to derive compositions of trigonometric and inverse trigonometric functions?

To prove: $$\sin({\arccos{x}})=\sqrt{1-x^2}$$ $$\cos{\arcsin{x}}=\sqrt{1-x^2}$$ $$\sin{\arctan{x}}=\frac{x}{\sqrt{1+x^2}}$$ $$\cos{\arctan{x}}=\frac{1}{\sqrt{1+x^2}}$$ ...
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2answers
154 views

Proof with functions and inverse - Spivak

How does he know that $f^{-1}$ is one-one? Doesn't he have to prove that? Or is he applying his first theorem in the chapter to $f$? That is $f$ is a function if and only if $f^{-1}$ is ...
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2answers
38 views

Explanation of this step in a modular arithmetic problem

The multiplicative inverse of $5$ is $7$, when using mod $34$. $$\begin{align*} 5\cdot x&=3\\[0.1in] 7\cdot 5\cdot x &=7\cdot 3\\[0.1in] 1\cdot x &=7\cdot 3\\[0.1in] x&=21 ...
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1answer
95 views

Inverse of a polynomial function

I want to find the inverse of $f(x)=\frac{3}{4}x^2-\frac{1}{4}x^3 $ when $0<x<2$. According to wolfram the answer is inverse I would like to know how can I find wolfram's inverse.
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0answers
41 views

Finding the inverse of $y = x\ln\left(\frac m{1-x}+1\right)$ where $0<x<1$ and $m > 0$

Find the inverse of $y = x\ln\left(\frac{m}{1-x}+1\right)$ where $0<x<1$ and $m > 0$. [Thanks for editing my question] :) Thanks,
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11answers
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Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
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1answer
34 views

Inverse of the function

Find the inverse of the function $y=5\times10^x$ Tried the inverse function by square rooting it but that also didnt work.
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4answers
166 views

Matrix inverse of $\left(A-I\right)$ given $A^{-1}$

I am wondering if the inverse of $$B = A-I$$ can be written in terms of $A^{-1}$ and/or $A$. I am able to accurately compute $A$ and $A^{-1}$, which are very large matrices. Is it possible to ...
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votes
1answer
50 views

Finding the inverse of a map from $CP^1$ to $S^2$

Given the map: $$f:CP^1 \to S^2\ ,\ f[z:w] = \left(\frac{2\mbox{Re}(w\bar{z})}{|w|^2+|z|^2},\frac{2\mbox{Im}(w\bar{z})}{|w|^2+|z|^2}, \frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$ How would I go about ...
2
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2answers
102 views

Intermediate Value Theorem, least upper bounds, Spivak

I have only taken an excerpt from the book from Spivak 3rd edition page 220 in his "Inverse Function" chapter. At the end of the 3rd paragraph, he says that Then $f$ takes on some value ...
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2answers
207 views

nonegative inverse eigenvalue problem

I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form: $$ \mathbf{M} = \begin{vmatrix} \mathbf{A} & \mathbf{b} \\ ...
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1answer
2k views

Finding matrix inverse by Gaussian Elimination With Partial Pivoting

Hello guys I am writing program to compute determinant(this part i already did) and Inverse matrix with GEPP. Here problem arises since i have completely no idea how to inverse Matrix using GEPP, i ...
0
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1answer
68 views

A interesting property of symmetric densities

Let $f(y)=f(-y)$ be a probability density function and furthermore let $f_0(y)=f(y-u)$ and $f_1(y)=f(y-v)$ be two densities based on $f(y)$ and $l(y)=\frac{f_1(y)}{f_0(y)}$ be the likelihood function. ...
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1answer
2k views

Inverse of a block matrix

I have a special case where $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ and: $X$ is non-singular $A$ is singular $B$ is full column rank $C$ is full row rank How do you ...
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1answer
40 views

Error in understanding the theorem about the invertibility of an element(coset) of a quotient ring

There's a theorem in Abstract Algebra which states that: An element of a quotient ring $\mathbb{Z}/\langle n \rangle$ or $\mathbb{Z_n}$ that is a coset $\overline{a}$ is invertible iff $a$ and $n$ ...
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3answers
263 views

Matrix Inverses

So in class we have been discussing matrix inverses and the quickest way that I know of is to get a matrix A, and put it side by side with the identity matrix, like $[A|I_{n}]$ and apply the ...
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0answers
38 views

Getting an inverse function

I have a cubic function $N_3(x) = a x^3 + b x^2 + c x + d$ which guaranteed is non-negative in each point on interval $x \in [0,1]$. I building an other function $N_4(x) = \int_0^x{N_3(t)dt}$. Sure ...
2
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2answers
77 views

Let $A,B$ be elements of $M_2(\mathbb{R})$. Give an example to show that $A+B$ can be invertible if $A,B$ are both non-invertible

The goal for this problem is to show that even if two matrices $A$ and $B$ are non-invertible, $A+B$ can be invertible. I tried to show this using a proof, but I ended up actually proving that this ...
2
votes
2answers
77 views

Let $A_{\alpha}$ be the $\alpha$-rotation matrix. Prove $A_{\alpha}^T = (A_{\alpha})^{-1}$

Let $A_{\alpha}$ be the alpha-rotation matrix. Prove $A_{\alpha}^T = (A_{\alpha})^{-1}$ In other words, prove $A_{\alpha}$ transpose = $A_{\alpha}$ inverse. First of all, what is a ...
3
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1answer
50 views

If $A$ is an $n \times n$ matrix such that $A^3 = O_{3}$, show that $I - A$ is invertible with inverse $I + A + A^2$

So this question is basically a proof. If $A$ is an $n \times n$ matrix (so square) which satisfies the condition $A^3 = O_{3}$ ($A^{3}$ gives the $3 \times 3$ zero matrix), then show that $(I - A)$ ...
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0answers
148 views

Sherman-Morrison inverse formula

I've read a paper in which the authors said that they use the "Sherman-Morrison inverse formula". While I know the Sherman-Morrison formula, I couldn't find anything about the inverse of said formula. ...
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0answers
84 views

General solution for $M^{\circ -1 }(y)=x $ when $g(x)e^{f(x) }=y$

Reading this question $e^{C/x }-1=D/(x + a) $, i found my self completely unable to do anything. This is much more hard for me than my easy exercises about Lambert $W$-function. So I probably need ...
4
votes
4answers
734 views

How to find inverse of the function $f(x)=\sin(x)\ln(x)$

My friend asked me to solve it, but I can't. If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$? I have no idea how to find the solution. I try to find ...
1
vote
2answers
129 views

inverse of laplace transform

How to compute this inverse Laplace transform ? $$\displaystyle{ \mathcal{L^{-1}} \left\{ \frac{1}{s(\exp(s)+1)} \right\} }$$ Thanks.
2
votes
2answers
1k views

What is the inverse z transform of 1/(z-1)^2?

I'd like to know how to calculate the inverse z transform of $\frac{1}{(z-1)^2}$ and the general case $\frac{1}{(z-a)^2}$