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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2
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3answers
27 views

Inverse Matrices and Infinite Series

Given that $C=I+A+A^2+A^3+ \ldots$ Prove that I-A is the inverse of $C$ Hint: Use the infinite series technique for finding inverse of a matrix. Now I know with an infinite geometric series with a ...
0
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1answer
13 views

Finding hermitian conjugate and inverse of a complex matrix

I have the following matrix: $$ F = [e^{i\frac{2\pi kl}{n}}]^{n-1}_{k,l=0} \in \mathbb{C}^{n,n} $$ for $n = 1,2,3,...,i$ I need to find $F^HF$ and $F^{-1}$ where $F^H$ is a hermitian conjugate ...
0
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0answers
25 views

Inverse Laplace transform of $\frac{1}{s} \frac{\sqrt{s}-1}{\sqrt{s}+1}$

I have been desperately trying to find the inverse laplace transform using the complex inversion formula for this question. $\frac{1}{s} \frac{\sqrt{s}-1}{\sqrt{s}+1}$ I have found it extremely ...
4
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5answers
44 views

Tool for expressing $x=f^{-1}(y)$ if $y=f(x)$ is given

I have many equations of nature - $y=ax^{12}+bx^5+5x^4+1$ For all these equations, I need to express x in terms of y. What tool or software would you recommend for this? I would much prefer to ...
2
votes
1answer
37 views

Calculate the inverse of $h(x)=f(2x)$

I have to calculate the inverse $f^{-1}(x)$ of $y=f(x)=2x-1$ and it is simple for this kinds of functions Let $x=f(y)=2y-1$ $x+1=2y$ $\displaystyle\frac{x+1}{2}=y$ We now have the inverse ...
3
votes
1answer
79 views
+50

If $f(x) = \sum \limits_{n=0}^{\infty} \frac{x^n}{2^{n(n-1)/2} n!}$ then $f^{-1}(f(x)-f(x-1))-\frac{x}{2}$ is bounded

For every $x>0$, let $$f(x) = \sum \limits_{n=0}^{\infty} \dfrac{x^n}{2^{n(n-1)/2} n!}.$$ Let $f^{-1}$ be the functional inverse of $f$. How to show there exists a positive real constant $C$ such ...
0
votes
0answers
30 views

Determine matrix from linear transformation

Let $T_{1}$ and $T_{2}$ be linear transformations given by $$T_{1}([x_{1}, x_{2}])=[3x_{1}+5x_{2}, 4x_{1}+7x_{2}]$$ $$T_{2}([x_{1}, x_{2}])=[2x_{1}+9x_{2}, x_{1}+5x_{2}]$$ Find a matrix A such that ...
0
votes
0answers
48 views

Can this be expressed by a contour integral?

Let $f(z)$ be a real entire function of the form $f(z) = a_1 z + a_2 z^2 + ...$ such that $0 < a_{n+1} < a_n$. Consider $g(x) = f^{-1}(f(x)-f(x-1))$ where $x$ is a positive real and $f^{-1}$ ...
0
votes
1answer
23 views

finding the inverse of a matrx

In order to decrypt a cipher text using hill cipher, we must first find the inverse matrix of a given matrix. From this link http://en.wikipedia.org/wiki/Hill_cipher, ...
1
vote
0answers
8 views

Speed of pseudo-inverse (with possibly ill-conditioned matrices)

I am computing the pseudo-inverse of several matrices of identical size $m \times n$ . However, computation (e.g. with the LAPACK pinv) seems to be much slower in some cases (5 to 10 times slower). ...
0
votes
1answer
19 views

Inverse Laplace transformation of (s^2-4s-2)/((s^2+2)^2)

I approached this problem as follow: $1.$ rewrote $(s^2-4s-2)$ into $(s-2)^2-6$ $2.$ Now break the function into 2 parts: $\frac{(s-2)^2}{(s^2+2)^2} + \frac{6}{(s^2+2)^2}$ the Laplace inverse ...
0
votes
0answers
42 views

What is the inverse of $f(x)=x^{x^x}$?

I'm curious to find the inverse of $ f(x)=x^{x^x} $ As an added extra, I'm already familiar with the Lambert Product Log function.
2
votes
0answers
29 views

Integral involving logarithm and inverse trigonometric function [closed]

$${\int\limits_0^1 {\frac{{\ln \left( {1 - x} \right)\ln \left( {1 + x} \right){{\ln }^2}x}}{{1 + x}}} },{\int\limits_0^1 {\frac{{\ln \left( {1 - x} \right)\ln \left( {1 + x} \right){{\ln }^2}x}}{{1 - ...
5
votes
2answers
89 views

Integral involving inverse of $x^x$

My brother gave me the following problem: Let $f:[1;\infty)\to[1;\infty)$ be such that for $x≥1$ we have $f(x)=y$ where $y$ is the unique solution of $y^y=x$. Then calculate: $$ \int_0^e f(e^x)dx $$ ...
1
vote
1answer
21 views

Schwarzian derivative of inverse function.

Let $\mathcal{D}$ denote the Schwarzian derivative. I have to prove that if $\mathcal{D}f(x)$ exists $\forall x$ then $\mathcal{D}f^{-1}$ exists $\forall x\in D_{f^{-1}}$ then find a formula. I tried ...
2
votes
0answers
44 views
+250

checking whether functions satisfy Inverse Function Theorem.

I've my exam tomorrow and this question is expected to come but donot know how to solve... Here's the INVERSE FUNCTION THEOREM stated in my notes: It says: Let $E\subseteq \mathbb R^n$ be open ...
1
vote
3answers
44 views

show that every rational number has one and only one multiplicative inverse

I am stumped and have no idea on how I prove this. I don't know what else to say. I am beyond lost.
1
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2answers
47 views

The inverse of AR structure correlation matrix / Kac-Murdock-Szeg ̈o matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots ...
1
vote
1answer
50 views

Proof that if $A$ is similar to $B$, then $B$ is similar to $A$

$A$ is similar to $B$ if there is an invertible matrix $S$ such that $B = S^{-1}AS$. Prove that if $A$ is similar to $B$, then $B$ is similar to $A$. So if $A$ is similar to $B$ then $B = ...
0
votes
2answers
42 views

Showing there is no invertible function $f: \mathbb{R} \to \mathbb{R}$

I'm wondering whether there is an invertible function $f: \mathbb{R} \to \mathbb{R}$ such that $f(-1)=0$, $f(0)=1$ and $f(1)=-1$. I think it's not but I'm missing a real proof. The easiest would be ...
0
votes
0answers
9 views

Sherman Morrison Formula for hermitian updates

I have a problem in which, in principle I can apply twice Sherman-Morrison formula but it seems to me that for this case, there should be a simpler solution so my question is "May the process ...
2
votes
1answer
43 views

How to get tangent of inverse of curve??

Ok so my question is. Let $ f(x)=(1/7)x^3+21x-1.$ and let y=g(x) be the inverse function of f. Determine all points on the graph of the inverse function g so that the tangent line is perpendicular to ...
0
votes
1answer
20 views

What will $A^+A$ and $A^gA$ actually or exactly get if $A$ is not invertible?

I know if $A$ is invertible then $A^{-1}$ is the inverse of $A$, and $AA^{-1}=A^{-1}A=I$. I just learnt the concept of Generalized inverses and Moore–Penrose pseudoinverse. For a matrix $A$ that is ...
0
votes
1answer
17 views

conditions for Gauss_jordan elimination with no pivoting

Please note that here is Gauss_jordan elimination which help us get inverse of A. I am wondering, is there any condition that it could work without pivoting? I try to prove this under column ...
0
votes
2answers
17 views

Help with proving matrix transpose and inverses.

I am really struggling with these type of proofs. Could someone please give me hints on how to prove them, I do know the basic properties of transpose and inverse. If $ \mathbf{A} $ is invertible and ...
0
votes
1answer
16 views

Quadratic Equation with Matrix [Prove Invertible]

The problem is: "The $2\times 2$ matrix A satisfies $A^2-4A-7I=0,$ where I is the $2\times 2$ identity matrix. Prove that A is invertible." The hint given is: "We are trying to a matrix that is ...
0
votes
2answers
29 views

How does one compute the inverse of the function $f$ that satisfies $f(3x-2) = x-1$? [closed]

The problem is: Given $f: \mathbb{R} \to \mathbb{R}$ such that $f(3x-2) = x-1$, find $f^{-1}(x)$. It would be great if you could help me on this one
0
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0answers
26 views

The converse of the inverse function theorem

The inverse function theorem: A continuously differentiable function $F=(F₁,...,F_{r+1})$ defined from an open set $U⊂ℝ^{r+1}$ into $ℝ^{r+1}$ is invertible at a point $z=(s₁,s₂,...,s_{r},s_{r+1})∈U$ ...
0
votes
0answers
13 views

Inverse of a 2x2 principal submatrix whose inverse is known

Let $H$ be a $n\times n$ symmetric positive definite matrix. What is the (computationally) quickest way to obtain $H_{ij}$, the $2\times 2$ matrix whose inverse is the principal submatrix of the ...
2
votes
1answer
47 views

Circle Equation Surjectivity

Consider the circular function $g:\mathbb{R}^{2} \to \mathbb{R}^{+}$, $g(x,y)=x^{2}+y^{2}$. Show that it is surjective and continuous. Note This post has been amended in accordance with the ...
1
vote
0answers
9 views

Inverse of a sum of cosh, or equivalently $U^{-1}$ for a normal coordinate transformation

I have an equation $\vec{R}_n= \sum\limits_{p=1}^N \vec{X}_p(t) cos(\frac{\pi p}{N+1}(n+\frac{1}{2}))$ of which i know the inverse is given by: $\vec{X}_p= \frac{1}{N+1} \sum\limits_{n=1}^N ...
0
votes
1answer
14 views

Reverse map for an equation .

I don't know this is actually reverse mapping or what but i have following equation. $$x = \tanh(a \cdot b ) + c $$ How do I solve for $a$? Does it has anything to do with inverse hyperbolic ...
1
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0answers
18 views

Definition of inverse binomial distribution

I am trying to succinctly define the inverse binomial distribution. Not the normal approximation, but the real thing, which will be discrete. So far I have this: $F^{-1}(\alpha;N,p) = k,\ \ s.t.\ \ ...
0
votes
3answers
39 views

Inverse of finite squared matrices.

I've usually used that given a square matrix $A$ with determinant $\det(A)\neq0$, then its inverse $A^{-1}$ is the matrix that meets: $$A^{-1}A=\mathbb{I}$$ and $$AA^{-1}=\mathbb{I}.$$ However, ...
1
vote
2answers
58 views

Prove $m=n$ of function $F:\mathbb{R}^n\to\mathbb{R}^m$ which has an inverse

Let $F:\mathbb{R}^n\to\mathbb{R}^m$ have an inverse function ${F^{-1}}:\mathbb{R}^m\to\mathbb{R}^n$ .If $F$ is differentiable at $a\in R^{n}$ and $F^{-1}$ is differentiable at $b=F(a)\in R^{m}$, ...
0
votes
2answers
54 views

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$.

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$. One thing to note is $a^{-1}\ne \large\frac{1}{a}$ (same goes for $b$) in this instance as there could be fields where this isn't ...
16
votes
6answers
2k views

What's the inverse operation of exponents?

You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the ...
0
votes
1answer
55 views

How to find the inverse of f?

$ f : A \rightarrow B $ where $ A = B = \left \{4,5,6,7 \right \} $ $ f = \left \{ (4,6),(5,5),(6,7),(7,5) \right \} $ Find $ f^{-1} $ I know how to find the inverse of $ f $ if it were ...
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votes
3answers
44 views

$A= \begin{pmatrix}1&0\\0&a\end{pmatrix}$ find a matrix such that $DA= I$ [closed]

Let $$A= \begin{pmatrix}1&0\\0&a\end{pmatrix}$$ a= any real number, find a matrix D such that $DA=I$ as a product of elementary matrices and find AD I tried it as [1 0 1 0] [0 a 0 1] but ...
2
votes
4answers
65 views

Why is $f(x)^{-1}$ used to denote the inverse of a function, and not its reciprocal?

Function notation says that any operations applied to a variable inside the parenthesis are applied to the variable before it enters the function, and anything applied to the function as a whole is ...
1
vote
1answer
18 views

Change of variable pdf inverse function

I've been given the following problem: $f(x,y) = e^{-(x+y)}$ on intervals $x \ge 0$ and $y \ge 0$, and $f(x,y) = 0$ otherwise. I'm also given that $Φ_1(x,y) = \frac{x}{y} = U$ and $Φ_2(x,y) = x + y = ...
0
votes
2answers
23 views

finding intervals on which f is a continuous inverse

I'm having trouble wrapping my head around this problem. I'm given a function f(x) - x + sinx and told to find all the intervals on which f has a continuous inverse. I honestly really have no idea ...
1
vote
2answers
32 views

Finding only first row in a matrix inverse

Let's say I have a somewhat large matrix $M$ and I need to find its inverse $M^{-1}$, but I only care about the first row in that inverse, what's the best algorithm to use to calculate just this row? ...
1
vote
2answers
20 views

Inverse of a function containing the ceiling function over the natural numbers

I am wondering if there exists an inverse function for $\lceil{e^{x}}\rceil$ over the natural numbers. I don't think it is a trivial task to derive an inverse function for a function containing a ...
0
votes
0answers
20 views

Using Cramer's Rule for inverse formula

Let $S$ be an $n\times n$ matrix with $\left|S\right| \neq 0$. Show that $S$ is invertible and that the inverse of $S$ is: $S^{-1}=\frac{1}{\left|S\right|}(-1)^{i+j} \left|S_{ij}^C\right|^T$ where ...
0
votes
1answer
30 views

How to find the inverse function of Euler's number?

Given: $f(x)= \dfrac{e^x}{1+9e^x}$ , what steps would I take to find its inverse? I tried following the steps on finding the inverse of a normal function but I keep getting one of the variables to ...
1
vote
0answers
38 views

What is the inverse kernel to this integral transform

What is the associated inverse kernel to the integral transform $T$ defined by \begin{align*} (Tf)(u) & = \int_{-\infty}^{0} \hat{f}(s)\exp((2i\pi+c)us)\ ds + \int_{0}^{+\infty} ...
2
votes
2answers
45 views

Question concerning Preimage

Let $f$ be the map from $\mathbb{R} \to \{a,b,c\}$ defined by \begin{equation} f(x)=\begin{cases} a &\text{if} \quad x>0 \\ b & \text{if} \quad x<0 \\ c &\text{if} \quad x=0 ...
0
votes
2answers
43 views

How to show that $AX=B$ has unique solution for invertible matrix $A$

If $A$ is an invertible $n \times n$ matrix, show that $AX=B$ has a unique solution for any $n \times k$ matrix $B$. I'm not sure where to start. What I have is that, if $A$ is invertible then ...
2
votes
1answer
53 views

Integration of a function containing inverse trigonometric functions

Q. $$\int \sin\left\{2\tan ^{-1}\left(\sqrt{\frac{3-x}{3+x}}\right)\right\}dx$$ $\implies$ $$\int \sin\left\{\sin ...