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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23 views

Help inverting a non-linear system of equations

I have a set of two equations like this $$ \gamma_3=\left(\frac{1}{\sqrt{1+2c_3^2+6c_4^2}}\right)^3 \left( \alpha_1\,c_3^3 + \alpha_2\,c_3c_4^2 + \alpha_3\,c_3c_4 + \alpha_4\,c_4\right)\\ ...
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47 views

Explanation on how is simplified expression $\frac{s^2+3s+3}{2s^2+7s+7}$

This is done in the solution of exercise in order to make it possible to do inverse Laplace transform. Though I am not sure how is that done, so here it is: ...
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79 views

Some questions about the pseudoinverse of a matrix

For every mxn-matrix A with real entries, there exist a unique nxm-matrix B, also with real entries, such that $$ABA = A$$ $$BAB = B$$ $$AB = (AB)^T$$ $$BA = (BA)^T$$ B is called the pseudoinverse ...
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37 views

Kalman filter innovation residual inversion

I'm trying to implement a Kalman filter in a computationally efficient way. The main issue is the inversion of the innovation residual: $$S=HPH^T+R$$ $$K=PH^TS^{-1}$$ My question is, can one assume ...
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36 views

Find Inverse Laplace Transform

I'm trying to calculate $$L_s^{-1}\left({\cfrac{2s+12}{s^2+9}}\right)=2L_s^{-1}\left({\cfrac{s+6}{s^2+9}}\right)$$ But I do not know how to go from here. I have noticed that the bottom does look ...
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27 views

Plotting the inverse of a function

The inverse of the function $y=2^x$ is $\bf (A)$ $y=\log_2x\quad{\bf (B)}\, y=-2^x\quad{\bf (C)}\,y=2^{-x}\quad{\bf (D)}\,y=x^2.$ Need help solving this problem and plotting it on a graph.
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How do I go about solving this derivative of inverse tangent?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$8\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=8\frac{1}{1+x^2}$$would ...
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1answer
31 views

How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
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24 views

Solving polynomial equation with 2 unknowns with Euclid (inverse of element in field)

Let $p(x) \in \mathbb Z_{5}[x]$, given by $p(x) = x^{3}+2x^{2}+1$ and let $I = <p(x)>$ be the ideal in $\mathbb Z_{5}[x]$ constructed by $p(x)$. Determine the inverse of $2x+3+I$ in $\mathbb F$ ...
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I need help finding the derivative of the inverse function.

So $$f(x)=\frac{x+1}{2x-1}$$ and $$g(x)$$ is an inverse of $$f(x)$$ I have the points on $f(x)$ of (2,1). So I know that $f(2)=1$, $g(1)=2$ and $g'(1)=\frac{1}{f'[g(1)]}$ so $g'(1)=\frac{1}{f'(2)}$ ...
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1answer
62 views

finding exact value of $\sec^{-1} 5$

Find the exact value of $\sec^{-1} 5$ (decimal answer). I know that $\sec^{-1}5=\cos^{-1}\dfrac{1}{5}$, but I don't know how to proceed from here. I drew a right triangle with sides $1$ and $5$ ...
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2answers
50 views

Working with groups. Finding the inverse of some $S_9$

I want to compute the inverse of: $\begin{pmatrix} 1&2&3&4&5&6&7&8&9\\3&2&1&6&5&9&4&8&7 \end{pmatrix}$ Sorry about alignment(they are ...
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4answers
100 views

Finding $\sin^{-1}(x)$ without using a calculator

I don't understand how to compute $\sin^{-1} (0.6293)$, to figure out the angle without using a calculator. I understand how to find the answer if I use a calculator but I don't understand the ...
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1answer
74 views

Inverse of a block 2x2 matrix

How to solve this type of problem: We've got a block 2x2 matrix : $$A=\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{bmatrix}$$ If matrices $A$ and $A_{22}$ are invertible, show that a ...
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1answer
51 views

Lipschitz continuity of inverse

Given a function f : $\mathbb{R}^n\to\mathbb{R}^m$, which is known to be Lipschitz continuous, can we say anything about the Lipschitz continuity of it's inverse function (in this case, the ...
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1answer
64 views

Matrix inverse and Change of basis

I have 2 Change of Basis Matrices $ S_{A,B} $ and $ S_{A,C}$ I want determinate $ S_{C,B} $ We know that $$ S_{A,B} S_{B,C} = S_{A,C} $$ $$ S_{B,C} = S_{A,B}^{-1} S_{A,C} $$ Now i'm quite not ...
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1answer
52 views

Blind deconvolution of a function convolved with itself

I have a function/vector $f$ that I know is the result of an unknown function $g$ convolved with itself: $f = g \ast g$ Is there any way to do a blind deconvolution on $f$ with this constraint?
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101 views

If $B$ the inverse matrix of $A^2$ show that the inverse of $A$ is $AB$

How do I continue from $A(AB) = (BA)A = I$ and how can we justify it if we don't know that $AB=BA$? EDIT: Also, how can we prove that if $AB=I$ then $ BA = I$? This is seperate from the question ...
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685 views

Is every self-inverse matrix diagonalizable?

If $A=A^{-1}$, is there always a matrix C such that $C^{-1}AC$ is a diagonal matrix (containing only -1 and 1 in the main diagonal) ? How can I check with PARI/GP, if a given matrix is ...
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0answers
47 views

Self-inverse matrices with integers with pairwise different absolut values.

Let A be a self-inverse matrix ($A=A^{-1}$) with integer values such that no two integers have the same absolut value. Let M be the maximum of the absolut values (maximum-norm) of A. Which M is the ...
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374 views

What can be said about a matrix which is both symmetric and orthogonal?

I tried to find matrices A, which are both orthogonal and symmetric, this means $A=A^{-1}=A^T$. I only found very special examples like I, -I or the matrix $$\begin{pmatrix} 0 &0& -1\\ ...
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28 views

Inverse of a special function

I have a function as follows, I would like to get the inverse of this function. What is the inverse of $f(x)$? $$ y = f(x) = - \log(1-[1-e^{-x^\alpha}]^\beta)$$ Is my answer correct? $$ f^{-1}(x) = ...
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1answer
42 views

Why is this finding inverse of a matrix by row operation not working?

the correct answer is $\begin{pmatrix} -5&3&-6\\-6&3&-7\\-2&1&-2 \end{pmatrix}$ So I think the mistake might be in the first two row operations but I see nothing?
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32 views

Inverse of Continuous Function on Closed Bounded Part of R. Why Bounded?

Consider the following proposition: Let $A$ be a closed bounded part of $\Bbb R$. Assume $f: A\rightarrow \Bbb R$ is a continuous injective function. Then $f^{-1}: f(A) \rightarrow A$ is also ...
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132 views

Multiplicative inverse of $0$

If I'm not mistaken, in a ring with identity, the additive identity cannot have a multiplicative inverse. I'm trying to prove this. Here's my attempt so far: Suppose $0\cdot a=1$ $$0\cdot a=1$$ ...
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44 views

What is wrong in the following calculation for the inverse of a matrix?

$\left[\begin{array}{ccc|ccc} 0 & 3 & 0 & 1 & 0 & 0\\ 4 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 5 & 0 & 0 & 1 \end{array}\right]$ ...
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Is there an efficient method to find all the self-inverse matrices with integers in a given range?

Given n and a range, for example [-10,10], is there an efficient method to find all nxn-matrices A with integers in the given range, which are self-inverse, that means the equation $A=A^{-1}$ holds ...
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125 views

Can a matrix A with the property $A=A^{-1}$ only have the eigenvalues -1 and 1?

If a matrix A has the property $A=A^{-1}$, are the only possible eigenvalues 1 and -1 ? How can the matrices with integer values and the property $A=A^{-1}$ be characterized ? I found out that if ...
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3answers
56 views

Is fractional inverse of a function a known thing?

I know there's fractional Fourier transform, fractional derivative, maybe some other transformations generalized from being discrete to continuous. Now I wonder if there's any way to generalize a ...
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25 views

Inverting complicated function (possibly using secant root finder)

So I have the following equation from the 2002 paper "A Rapid Hierarchical Rendering Technique for Translucent Materials" http://graphics.ucsd.edu/~henrik/papers/fast_bssrdf/fast_bssrdf.pdf Here is ...
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is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
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1answer
39 views

Inverse Laplace Transform of the following complicated form

What would be the inverse laplace transform of the following: I mean I want to solve this: $$ \large \mathcal {L^{-1} [ \mathcal {L}[{sin(at+b)}] . \mathcal{L} [{e^{xt}}] . e^{cs}}] = ? $$
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1answer
19 views

Inverse matrix - transformation

I am finding inverse matrix $A^{-1}$ and I was given hint that I could firstly find inverse matrix to matrix B which is transformed from A. $$A=\begin{pmatrix}1 &3 & 9& 27\\3 & 3 & ...
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40 views

How to integrate the inverse of sin

How does one integrate $\int\ {\sin^{-1}(x)}$, using integration by parts, where: $$ u={\sin^{-1}}, du=\frac{1}{\sqrt{1-x^2}},dv=dx, v=x ? $$ This is a partial solution, and I do not quite ...
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64 views

Why rotating a function around line $y=x$ gives an inverse of this function?

So I'm trying to read through a book on calculus on my own and there is a statement that if we have a graph of some function $y=f(x)$ and this is an injective function, then rotating it around the ...
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55 views

How do I find the inverse of $e \bmod (p-1)(q-1)$?

I'm trying to find this inverse modulo to set up a solution for an RSA cipher. I haven't the slightest how to go about this. When I looked up the formula for such a question, it states: $$ d \equiv ...
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55 views

Domain of arctan(1/x)

I had this as part of a question in an exam. And, I reasoned, even when it's arctan(1/0) (undefined), it is pi/2. And, so I said, domain belongs to all Real Numbers. Why isn't it this
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45 views

Find the poles and residues in an awkward Laplace inversion

Assume that part c) has been proved and ignore parts c) & d). To invert the Laplace transform we would do $\displaystyle u(x,t)=\frac{1}{2\pi ...
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83 views

What is the inverse function of $x-\log(\log(x))$?

What is the inverse function of $f(x)=x-\log(\log(x))$? If we restrict the domain to e.g. $x\in[2,+\infty[$, the function should have an inverse, but I am unable to compute it.
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63 views

Method for Finding Matrix-Inverse Through Gauss-Jordan?

When trying to find the inverse of the n$\times$n matrix $A$, one way of going about it is by solving $AX=I$, wherein $I$ is the n$\times$n identity matrix, and $X$ is some n$\times$n matrix which is ...
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109 views

Contradiction in inverse Laplace transform problem with Mellin's inverse formula?

Let say we have to solve a given differential equation $$ty''+y'+ty=0$$ $$y(0)=1,\ y'(0)=0$$ (which is Bessel equation with the solution $y=J_0 (t)$, of course) with the Laplace transform. Then we ...
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0answers
51 views

Moore-Penrose Pseudo-inverse of a matrix on adding 1 new row/column

Given that I know the pseudo-inverse of a matrix(not necessarily a square matrix), how to calculate the pseudo-inverse of the matrix I get by adding a single row/column to the original matrix? i.e, ...
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75 views

If Q is a p-Sylow-Group of H there is a p-Sylow-Group P of G with $\phi(P)=Q$ while $\phi:G\rightarrow H$ epimorphism

Let G be a finite group and $\phi: G \rightarrow H$ a group-epimorphism. Proof: If $Q\in Syl_p(H)$ there is a $P\in Syl_p(G)$ with $Q=\phi(P)$.
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1answer
19 views

Finding the marginal density function of Y

Okay, the question is like this: $f_{x}(x) = xe^{-x^2/2}$ for all $x>0$ and $Y = \ln X$, find the density of $Y$. I don't understand a particular step of this problem. First they start for $x ...
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0answers
44 views

matrix inverse and limit

I would like to get a better understanding of limits and matrix inverses, specifically the relationship between: $\lim_{k\rightarrow \infty}(\mathbf{A}^{-1})$ and $(\lim_{k\rightarrow ...
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30 views

determinant and trace of a huge positive definite matrix

I have a problem to compute the determinant and the trace of inverse matrix: $det(\Gamma^{-1}+I_n⊗\Phi^T\Phi)$ and $tr[(\Gamma^{-1}+I_n⊗\Phi^T\Phi)^{-1}]$ where $\Gamma$ is a huge positive definite ...
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59 views

Continuity of the inverse matrix function

For a differentiation module I am taking one of the exercises (not homework) asks: Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $det(A) \neq 0$ is open. Let $A^{-1}$ be the ...
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7answers
627 views

How to find the inverse of the function?

$$f(x)=\frac{x+2}{5x-1}$$ Answer: $$f^{-1}(x)=\frac{x+2}{5x-1}$$ Can one of you explain how the inverse is the same exact thing as the original equation?
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1answer
28 views

local invertibility does not imply global invertibility

What is an example of a smooth function with continuous derivatives, that is locally invertible but not globally, and the reason for that is not injectivity. My first idea was $f:\mathbb{R}^{2}\to ...
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4answers
43 views

Set of all matrices with determinant 0, non-zero

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes ...