Tagged Questions

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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0answers
9 views

Find a right inverse of a map with gauss brackets.

I am having a composition of two maps: $$ f:\mathbb{R}->\mathbb{R_0^+},f(x)=x^2 $$ $$ g:\mathbb{R_0^+}->\mathbb{\mathbb{N}},g(x)=\lfloor x\rfloor $$ $$h=g\circ f:\mathbb{R}->\mathbb{N_0}$$ ...
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0answers
4 views

inverse hyperbolic function of a complex argument

It is not too hard to prove that $f(z)=\cosh z$ is a bijection from $$\def\C{{\Bbb C}}D=\{\,z=x+iy\in\C\mid 0<y<\pi\,\}$$ to $$R=\{\,w=u+iv\in\C\mid v\ne0\,\}\cup\{\,w\in\C\mid ...
3
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1answer
22 views

When is Block-Partitioned Matrix Invertible?

Suppose I have a block partitioned matrix \begin{equation} \begin{bmatrix} \mathbf{X}_1^{\top}\mathbf{X}_1 & \mathbf{X}_1^{\top}\mathbf{X}_2 \\ \mathbf{X}_2^{\top}\mathbf{X}_1 & ...
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1answer
37 views

Linear Algebra - inverse matrices $AB^{-1} = B^{-1}A$? [on hold]

in what condition is matrices $AB^{-1} = B^{-1}A$ true?
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1answer
20 views

What is a quickest way to find inverses of functions of two variables?

Suppose I have $X_1 = aY_1+bY_2$ $X_2 = cY_1+dY_2$ How is the quickest (or most efficient way ) to find the inverse functions? The current way I am doing it is attempting to solve for Y1 in the ...
3
votes
3answers
34 views

Inverse of a Function exists iff Function is bijective

How to mathematically prove that inverse of a function, let's say, $f^{-1}$, exists, if and only if $f$ is bijective? I know how to prove it using diagrams but I'm looking for a rather mathematical ...
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3answers
28 views

Multiplicative inverse

What is the multiplicative inverse of 7 modulo 11? Is this correct: $$7 = 11(0) +7$$ $$11 = 7(1) +4$$ $$7 = 4(1) +3$$ $$4 = 3(1) +1$$ We then take 3 equations: $$4 = 11 + 7(-1)$$ $$3 = 7 + 4(-1)$$ ...
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0answers
22 views

Finding the inverse of an integral

I'm looking for a computational approach here, since I don't think there is a closed-form solution. I have the following: $$ s(x) = \rho + \int_{\rho}^{x} \sqrt{ 1 + (\alpha \cos t - k)^2 } \, dt $$ ...
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0answers
8 views

Sum of Inverse Partitioned Matrix

Given a matrix $X(n\times p)$, divide $X$ by row into $K$ parts, $X_1, X_2...X_K$ each of which consists of the same amount of row vectors in $X$ as its own row vectors. Now consider ...
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1answer
24 views

Using Derivatives and Tangent Line to Find Area

Let $(a, b)$ be an arbitrary point on the graph of $y=\frac1x$ ($x>0$). Prove that the area of the triangle formed by the tangent line through $(a,b)$ and the coordinate axes is $2$ square units. ...
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1answer
56 views

Need help to find the inverse of a mapping implicitly

Let $f: \mathbb R^3\to \mathbb R^3$ be the linear mapping which reflects $x$ over the plane $x1+x2+x3=0$. You are given that the standard matrix for $f$ is: ...
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0answers
24 views

What's the algebraic definition of the inverse of a function?

I have a function $f(x)$, using the logic it's relatively easy to formally define what the inverse should be like, relatively to domain and codomain elements, especially using the surjective and ...
3
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1answer
52 views

Explicit formula for inverse matrix elements

Let $A$ be an $n \times n$ invertible matrix with \begin{align} \left(\begin{array}{ccc} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} ...
1
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1answer
109 views

A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding $A^{-1}$

The question is: A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding A^-1. I have looked at other similar questions on this site: 1. Here 2. and Here But they use ...
5
votes
1answer
129 views

Statement about $(I-A)^{-1}$ matrices

Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix. Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is such a diagonal ...
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4answers
58 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
14 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
31 views

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

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
votes
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
38 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 ...
4
votes
1answer
103 views

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 ...
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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
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0answers
48 views

Can this be expressed by a contour integral? [on hold]

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
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0answers
9 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
21 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 ...
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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
30 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
93 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
23 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 ...
3
votes
1answer
71 views

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 ...
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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
51 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
45 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
18 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 ...
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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
17 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
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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 $z=(s₁,s₂,...,s_{r},s_{r+1})∈U$ ...
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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 ...
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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
vote
0answers
20 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
61 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 $b=F(a)\in R^{m}$, ...