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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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}\| & = ...
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Self inverting Rings

Would it be possible for a ring to have elements that are their own additive inverses? What I mean is, would it be possible to have a ring $K$ of mathematical objects $A$ such that: $$A+A=i,\;\forall ...
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Finding inverse using logs

$$ x = \left(\frac{4^y}{-2}\right)^{\frac{1}{3}} $$ i have correct answer of $\:y=\log(4)-2x^3$ i'm lost on steps to obtain the answer. i tried the ...
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“Self invertible” group

Let there be an Abelian group with a binary operation $\ast$ on a set $S$. Let such a group respect the following propriety: $$ (X\ast Y)\ast Y = X$$ For any $X$ and $Y$ in $S$. I realize that by ...
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Problem inverting a function

I have this function: $$v(t)=\sqrt{\frac F c} \tanh \left(\frac{\sqrt{Fc}}{m} t \right)$$ I can visually see that t=6.3 when v=27.8, so why don't I get t=6.3 upon putting v=27.8 in this supposedly ...
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For any continuous function f(x), how can I split up the function and restrict the domain to find an inverse?

I want to know everything there is to know about inverses for curiosity's sake. I am totally fine finding the inverse of a function where each x maps to a unique y coordinate, but when we get to ...
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Inverse functions determination by integral

From "Inverse functions and differentiation": Integrating this relationship gives $$ f^{-1}(x)=\int\frac{1}{f'(f^{-1}(x))}\,dx + c. $$ This is only useful if the integral exists. ...
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Finding the inverse of trig functions

I'm supposed to find the inverse of $$f(x) = \cos(x)+x$$ I usually just substitute $x$ for $y$ and then re-arrange. What do I do in this scenario?
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multiplicative inverse in factor ring

If I need to find the multiplicative inverse of an element in some $T[x]/(m)$ factor ring, do I need to solve a diophantine equation to get the solution? Let the element be $f$. Then $fu \equiv 1$ ...
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Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
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Is there a pseudo inverse $X$ such that $ABX=A$?

Question The title pretty much sums it up. I need to find a matrix $X$ such that: $A B X = A$, with $A\in R^{n\times n}$, $\text{rank}(A)=n$, $B\in \mathbb{R}^{n\times m}$ given. The matrix $X$ ...
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Derivatives of component inverse functions

I might have missed the point of the following questions. Anyone kindly give a suggestion? Let $f:\mathbb{R}_\mathbf{x}^3\to\mathbb{R}_\mathbf{y}^3$ and ...
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Number of configurations in a constrained nested loops and configuration back from serial

Consider 4 counters looping the digits 0, 1, 2 to form the various "configurations", like in : ...
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Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as with SVD?

Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as using SVD?
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Extended Euclidean Algorithm for Modular Inverse

I'm currently learning how to find the inverse of a modulo with the Extended Euclid Algorithm and I stumbled upon a problem when finding an inverse when the $m>p$ as for $m \equiv 1 \pmod{p}$ For ...
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How to invert a matrix

I would like to disprove the following claim, that seems false to me, finding a counterexample. Let $\mathbf{A}, \mathbf{B} \in \mathbb{R}^{n \times k}$, for $k < n$. Let us assume that $rk(A) ...
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34 views

Nonsingular block matrix

Let us consider a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ and the block partitioning $$ \mathbf{A} = \begin{pmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & ...
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Inverse modulo function

How can we calculate the inverse of a modulo function, now I have a problem given me $f(n)=(18n+18)\mod29$, need find inverse of $f(n)$ ? how is the process to do it?
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How do I calculate the inverse of these matrices?

In learning how to rotate vertices about an arbitrary axis in 3D space, I came across the following matrices, which I need to calculate the inverse of to properly "undo" any rotation caused by them: ...
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Why is a matrix $A$ that fulfils $AA^t = I$ invertible?

Given a square matrix $A$ that fulfils $$AA^t = I$$ Justify why must $A$ be invertible. The answer, according to my book, is simply $$AA^t = I$$ $$A^t = A^{-1}$$ I don't ...
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calculating the inverse of an upper triangular matrix

I've been given this hint that if we have an upper triangular n-by-n matrix $R=\begin{bmatrix} \alpha & v^{T}\\ 0 & S \\ \end{bmatrix}$ then $C=(R^{T}R)^{-1}$ can be computed as ...
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67 views

number of flops required to invert a matrix

I have a n-by-n upper triangular matrix R and I can calculate it's inverse by back substitution. I cannot make myself see why it needs $O(n^{3}/3)$ flops to do so. Can you explain?
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If a matrix is not invertible, is it still possible to find a left and/or right inverse?

I was recently asked to find the right inverse of some matrixes. I found that all three of them were invertible, so it was just a matter of finding their inverses, which would be exactly the same as ...
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About inverse matrixes

I've been reading about invertible matrixes. I have a few questions: One theorem says The rank of an invertible matrix of size $n$ is $n$. So, is it safe to say that all invertible matrixes ...
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Inverse $2^{18}$ in GF(23) without extended euclidean algorithm

I have a little question about the calculation of the inverse of $2^{18} \mod\ 23$. I have the solution of this: $$ \text{The inverse of $2^{18}$ is $2^{-18}$. The modulus in the exponent is ...
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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. ...
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How to prove that a matrix inverse is invertible?

First off, I'm trying to prove that $(A^{-1})^{-1} = A$, but in my proof, I assume that $A^{-1}$ is invertible. I'd like to see or do a proof that $A^{-1}$ must be non-singular, but I'm stuck at ...
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inverse function as a differential equation

let be the functional equation $$ f^{-1} (x)=g(x) $$ (1) where $ g(x)$ is known but $ f(x) $ is not my question is if i can rewrite (1) as the solution of a certain nonlinear differential equation ...
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How do I find the inverse of this exponential function?

$x=-3(3^{-x})+9$ I know the steps up until a certain point. $x=-3(3^{-y})+9$ $x-9=-3(3^{-y})$ $\frac{(x-9)}{-3} = 3^y$ $ln (\frac{x-9}{-3}) = -y * ln 3$ Not sure what to do from here. I know I ...
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the rank of a matrix and its inverse are always equal

I had a true or false quiz in a linear algebra course, one of the statements read the rank of a matrix and its inverse are ...
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Conditions for a matrix to be invertible

Let $n \geq m$ and let $C$ be a $n \times m$ full rank matrix, that is $rank(C) =m$. Considering that $D$ is a diagonal positive semidefinite matrix, under which conditions is the $ m \times m$ matrix ...
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67 views

What does it mean when a matrix is to the (-1/2) power?

I'm reading a machine learning paper that uses a form of matrix normalization called symmetric divisive; given a matrix A and a diagonal matrix D derived from A, we define $$N=D^{-1/2}AD^{-1/2}$$ I am ...
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Yet another inverse function to calculate

Is it possible to evaluate the inverse of this function, in order to obtain for each $y\in\mathbb R^+$ an explicit value of $f^{-1}(y)$? Thanks in advance! ...
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Hard time with Derivatives of Inverse Functions

I'm having a really hard time with this question I keep googling for advice but can't find anything solid that's similar! Please help. I'm not sure if I should derive first or find the inverse first? ...
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Where exactly is the following process incorrect to yield an impossible answer

I was playing with my calculator and found some strange phenomena. $\cos(\tan(\tan(\tan(\pi/4)))) = 0.75686700166$ Verify here Now when we apply some inverses, then $\tan(\tan(\tan(\pi/4))) = ...
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Linear Algebra Review Questions

So I have a test on Monday and my professor posted a couple of non-graded review questions that she said we should look over. Anyhow, I have a couple of questions that I'd like answered if that's ...
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Determining the values of k for which the Matrix A has an inverse

I've been given this question in class, with the 3x3 matrix: 2 1 0 1 2 1 0 -3 k My job here is to find the values of k for which this matrix has an ...
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From concrete mathematics problem 4.35

From concrete mathematics problem 4.35. Let $I(m,n)$ be function that satisfies the relation $$I(m,n)m+I(n,m)n=\gcd(m,n)$$ when $m,n\in \Bbb Z^+$ with $m\neq n$. Thus, $I(m,n)=m′$ and $I(n,m)=n′$ ...
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Tangent line of the inverse function of $y = e^x + x$

I've been sitting on this problem for a while, hopefully you guys could give me a lead on what the hell is going on :) Let $f(x) = e^x + x$ Find the tangent line to $f^{-1}(y)$ (the inverse ...
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How to find inverse of…

How do you find the inverse of the equation in the form $y= b^{x-h} +K$ for example: $y=2^{x-4} +6$ I already know that the inverse of $b^x$ is $\log_bx$ but how do you find with the $H$ and $K$ ...
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Find the function $h(x) = g(2g^{-1}(x))$

Show that the function $g(x) = x^4 + x^3 + 1$ is one-to-one on [0, 2]. In addition, for the function $h(x) = g(2g^{-1}(x))$, find h′(3). For the first part, I manage to prove that g(x) is increasing ...
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41 views

an inverse of m with respect to n

From concrete mathematics problem 4.35. Let $I(m,n)$ be function that satisfies the relation $$ I(m,n)m + I(n,m)n = \gcd(m,n),$$ when $m,n \in \mathbb{Z}^+$ with $m ≠ n$. Thus, $I(m,n) = m'$ and ...
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How do I find $(f^{-1})'(a)$? [closed]

if $$f(x) = 3x^3 + 3x^2 + 6x + 9 $$ $$a = 9$$ and also $$f(x) = 2x^3 + 3\sin x + 3\cos x$$ $$a = 3$$ I know I have to find the inverse but I think I’m getting overly complicated answers and my ...
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Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
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Optimization that involves inverse operation.

$\newcommand{\diag}{\operatorname{diag}}$ I have the following optimization problem: \begin{align} \mathop{\arg\min}_\beta & \frac{1}{2} a' [ M + \diag( \beta ) \otimes I_d ]^{-1} a + ...
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Linear Growth Model

I have a problem where I have been given that $r(t)=at+b, 0 \leq t \leq \frac{100-b}{a}$. I have then been asked to find $t(r)$. Is this simply finding the inverse of $r(t)$?
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The rank of general inverse of $A$ times $A$?

Supposing $X$ is the general inverse of $A$, that $AXA = A$. Then $XA$ is idempotent, that is $(XA)(XA) = XA$. Why is the rank of $XA$ equal to the rank of $A$ ? Thanks.
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Inversion of Matrix

What is the inverse of the following (n x n)-matrix? $$ \begin{bmatrix} 2 &-1 &0 &0 &... &0 &0 \\ -1 &2 &-1 &0 &... &0 &0 \\ 0 &-1 &2 &-1 ...
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How can I find the inverse of this function? [closed]

Can anyone help me find the inverse of this function? $$y=\frac{x}{2}-\frac{x^2}{16}$$
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Find the inverse function of a function relating to limited exponential sum

The function is given out as: $$y = 4x + {x^m} + {x^{ - m}},where{\text{ 0 < m}} \leqslant {\text{1, 0 < }}y < 6;$$ Closed form will be highly appreciate,but approximate results is also ...