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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30 views

what is the name of this matrix? does it have any special characteristics?

does anyone know the name of this matrix or if it has any special characteristics or how to calculate its inverse efficiently e.g. in a closed-form? [ \begin{array}{llllll} ...
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1answer
80 views

What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$ \frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0 $$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
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1answer
21 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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43 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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74 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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1answer
33 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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34 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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1answer
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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49 views

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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21 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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70 views

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
60 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
42 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
96 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
31 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
28 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
52 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
32 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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4answers
99 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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581 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
43 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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2answers
357 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
41 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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1answer
22 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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106 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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41 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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0answers
41 views

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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3answers
124 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
50 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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1answer
21 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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24 views

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
37 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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1answer
37 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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3answers
46 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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1answer
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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2answers
46 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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42 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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1answer
81 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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2answers
39 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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0answers
38 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 ...
3
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0answers
44 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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3answers
68 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
18 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 ...
2
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0answers
33 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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0answers
27 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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2answers
40 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 ...
2
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7answers
598 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?