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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Finding the Inverse of a 5th order Function

I need to get the inverse of that function. Can I get some help please? Thanks! $$ f(x) = -x^5-2x+2 $$
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861 views

left inverse is not equal to right inverse [duplicate]

Is it possible to have a function which has both left and right inverse but they are unequal ? A left inverse means the function should be one-to-one whereas a right inverse means the function should ...
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111 views

A formula for n-derivative of the inverse of a function?

Let $y=f^{-1}(x)$. As we know: \begin{align} \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{{f}'(y)} \end{align} Thereof we have: \begin{align} \frac{\mathrm{d^2} y}{\mathrm{d} ...
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216 views

Find the domain and range of $y=\cos^{-1} \sqrt{1-x}$

Find the domain and range of $y=\cos^{-1}\sqrt{1-x}$. Can someone please help me with question above, as to how it's done? Thanks. I am unfamiliar with what you do when there is a square root.
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81 views

Nilpotent matrices and inverses

Can somebody give me a hint for showing that: The matrix $A+I$ is invertible if there is an integer $k\gt 0$ so that $A^k=0$.
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159 views

How to find the inverse function of $f: \mathbb R^2\to\mathbb R^2$?

How to find the inverse function of $f: \mathbb R^2\to\mathbb R^2$? for example, $u= x+2y, v=xe^y$, how to find the inverse?
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503 views

Sherman–Morrison–Woodbury formula question

I asked a question earlier on Inverse of (A+B) and (A+BCD)? I got a very good hint on using the Sherman–Morrison–Woodbury formula. I used R to code the idea and using some arbitrary matrices, I ...
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221 views

Red Inverse, Blue Inverse, Left Inverse, Right Inverse

1.) Prove that $f$ has a left inverse if and only if $f$ is injective. 2.) Prove that $f$ has a right inverse if and only if $f$ is surjective. My first attempt for 1.): $$f\text{ is ...
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205 views

Inverse of sparse matrix is not generally sparse

I have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices). I encountered several times the web pages which states that the ...
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1answer
31 views

Poins on Inverse functions

I have tried to do this but, I don't know what to do. They have given me a function y=h(x) that passes through point (3,2) and (4,5). If the function h has a inverse called j, what points must be on ...
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88 views

Does $\exp(\ln(I+A))=I+A$ when $\|A\|<1$?

For matrices, I know certain equalities like $e^{A+B}=e^Ae^B$ aren't always true. I'm curious, do $\exp$ and $\ln$ serve as inverses? I saw earlier that if $\|A\|<1$, then $\ln(I+A)$ converges. My ...
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Find inverse for the closed-form expression of linear recurrence relation

I am trying to find an inverse of the following formula: $$ a_n=\frac{2+\sqrt{6}}{4}(1+\sqrt{6})^n+\frac{2-\sqrt{6}}{4}(1-\sqrt{6})^n $$ This formula is a closed-form expression of a linear ...
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226 views

Analytical pseudo inverse of rank deficient matrix by QR decomposition

Given an overdetermined real linear system $Ax = b$, with $m$ equations, $n$ variables, and some unknown elements in both $A$ and $b$. I need to know what constraints must be imposed on those unknown ...
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108 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. ...
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285 views

Intuition/Understanding of Inverse and Determinants

This is not homework, but extends from a proof in my book. EDIT We're given an $m \times m$ nonsingular matrix $B$. According to the definition of an inverse, we can calculate each element of a ...
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49 views

Trying to create fitness function

I'm trying to create an idea fitness function using an inverse parabola but I'm struggling with the math. Basically what I have is: ...
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52 views

Sketching the Inverse of a Function [closed]

I can't figure out how to sketch $f^{-1}$ if $f$ has the domain of $[-3, 3]$.
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164 views

Find the matrix $S$ of stretch by a factor of $3$.

All mappings are from $\mathbb{R}^2$ to $\mathbb{R}^2$. Find the matrix $S$ of stretch by a factor of $3$ in the $y$-direction and the matrix $S^{-1}$. So the matrix $S$ is a $2\times2$ matrix. So ...
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1answer
64 views

Monoid with inversion

Is there a name for monoid with operation $a\mapsto a^{-1}$ conforming the equations $(a^{-1})^{-1}=a$ and $(b\cdot a)^{-1} = a^{-1}\cdot b^{-1}$? (with no requirement that $a^{-1}\cdot a$ would be ...
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46 views

A question about unital Algebra over a Field

Let $F$ be a field and $(K,\bullet)$ be a unital $F$-algebra. For each $v \in K$ let $S(v)$ be the set of all $a \in F$ satisfying the condition that $v -a.1_K $ does not have an inverse with respect ...
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Prove that if $AB$ is invertible then $B$ is invertible.

I know this proof is short but a bit tricky. So I suppose that $AB$ is invertible then $(AB)^{-1}$ exists. We also know $(AB)^{-1}=B^{-1}A^{-1}$. If we let $C=(B^{-1}A^{-1}A)$ then by the invertible ...
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1answer
67 views

Can this matrix inverse be re-written?

I have a matrix inverse of the form $(\mathbf{AB}+\mathbf{C})^{-1}$, where each matrix is $2\times 2$ and each of the subelements below are known a priori. $$ \left( \left[ \begin{array}{cc} ...
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444 views

Derivative of Standard Normal Inverse

How can I calculate the derivative of the standard normal inverse. I think the derivative of $\Phi^{-1}(x)$ is $$\frac{1}{\phi(\Phi^{-1}(x))}.$$ I would like to know how to find the derivative of ...
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267 views

Is the inverse of a diagonal dominant matrix also diagonal dominant?

Is it guaranteed that the inverse of a diagonal dominant matrix, whose elements are all nonnegative, is also diagonal dominant? In my specific problem, I have a diagonal dominant complex matrix ...
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525 views

Pseudo inverse of a product of two matrices with different rank

Let $V$ be an $n \times n$ symmetric, positive definite matrix (of rank $n$). Let $X$ be an $n \times p$ matrix of rank $p$. Define $A^- = (A^\top A)^{-1} A^\top$ as the pseudo inverse of $A$ when ...
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108 views

Deconvolution of a convolution product with $Ax\ /\ (x^2+l^2)^{3/2}$

This is not a homework, and I have no idea whether it could be one. It is only a request for help, as I do not have any experience using Fourier transform. The origin of the problem is from physics. ...
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1answer
55 views

Can I use the pseudoinverse of a Jacobian like I think I can?

I need to compute the Jacobian for a transformation that maps parameters $p_1,...,p_n \to q_1,...,q_m$, $n\neq m$. For this, I need to compute the derivatives $\frac{\partial p_i}{\partial q_j}$. ...
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2answers
92 views

A bijective mapping from $\mathbb N^k$ to $\mathbb N$?

Having $k$ numbers $N_i\in\mathbb{N}$, I'm looking for a bijective mapping $f:\mathbb{N}\times\ldots\times\mathbb{N}\rightarrow\mathbb{N}$ So that ...
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1answer
248 views

Prove the inverse of the Hilbert matrix has integer entries [duplicate]

$1 \frac{1}{2} ... \frac{1}{n}$ $\frac{1}{2} \frac{1}{3} ... \frac{1}{n+1}$ $.$ $.$ $.$ $\frac{1}{n} \frac{1}{n+1} ... \frac{1}{2n-1}$ Does the inverse of this matrix ...
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56 views

One-to-one functions between vectors of integers and integers, with easily computable inverses

I'm trying to find functions that fit certain criteria. I'm not sure if such functions even exist. The function I'm trying to find would take vectors of arbitrary integers for the input and would ...
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237 views

How to reverse a range of numbers

if I have a range of floating point numbers, specifically this: min: 0.5 max: 1.5 How do I reverse the order of a selected place in the range? For instance if I ...
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1answer
92 views

Derivative of the inverse of $y=(a+bx)e^{cx}$

I need to solve for the 1st derivative of the inverse of $y=(a+bx)e^{cx}$ but my calculus is a bit rusty. I know that to get the inverse function, I would have to use the Lambert W method but I think ...
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50 views

Creating Inverse Function with Certain Characteristics

Given the following: $D : \in \Re [-n, n]$ $R: \in \Re[-\frac\pi4, \frac\pi4]$ The curve of the function should be completely smooth, and can be undefined outside the given Domain. The graph should ...
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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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317 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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95 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
125 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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236 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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100 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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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
228 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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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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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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74 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 ...
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340 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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429 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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96 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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150 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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53 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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83 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 ...