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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Geometric interpretation of inverse complex function?

Function $f\colon\mathbb{R}\to\mathbb{R}$ and its inverse $f^{-1}$ are symmetric over line $y=x$. It's easy to imagine inverse of real function, we just have to "flip" the plot over $y=x$. But what ...
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0answers
24 views

Solve the system of trigonometric equetions, inverse kinematics

I am trying to do inverse kinematics for some mechanical system. After applying Neton-Euler method following equations were obtained: $$F_x = k_f w_l\sin(\beta_l) + k_f w_r\sin(\beta_r)$$ $$F_y = k_f ...
0
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0answers
79 views

If $f^{-1}(x)$ is continuous, is $f(x)$ also continuous?

Let $f:\mathbb{R}\mapsto\mathbb{R}$ be a one-to-one function with $f(\mathbb{R})=\mathbb{R}$. If $f^{-1}(x)$ is continuous $\forall x\in\mathbb{R}$, prove or disprove that $f(x)$ is continuous ...
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1answer
33 views

Inverse sum representation of sine

The other day I was playing with functions of the form $$ f(x) = \frac{1}{\frac{1}{a_0(x-b_0)} + \frac{1}{a_1(x-b_1)} + \cdots + \frac{1}{a_n(x-b_n)}} $$ and I found particularly that $$ ...
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1answer
23 views

Cayley transform a matrix that is invertible when added to the identity

Let A be an nxn matrix such that (I+A) is invertible. I need to prove that the Cayley Transform of A, denoted by $A^c$, is such that $(I+A^c)$ is invertible. The Cayley Transform is defined as ...
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30 views

Geometric Interpretation of Determinant of an Inverse Matrix

The $\mathbf{A}$ be an $n\times n$ full rank matrix. Then, the (signed) volume enclosed by the rows (or columns) of $\mathbf{A}$ is equal to $\det(\mathbf{A})$. My question is, what is a geometric ...
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1answer
126 views

Proof of the Inverse Function Theorem using the Contraction Mapping Principle.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
3
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1answer
45 views

Reciprocal of the reciprocal of zero

By straightforward evaluation, $$(0^{-1})^{-1}=(NaN)^{-1}=NaN$$ where $\frac{1}{0}$ is taken to equal $NaN$ (not a number), or undefined or indeterminate. However, the laws of exponents state that ...
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1answer
18 views

Is the condition “the inverse image of a closed base set is closed?” sufficient for continuity?

Let's say you have a function $f:X \to Y$, where $X$ and $Y$ have topologies. The set $C$ forms a closed base for $Y$. If for every $c \in C$, $f^{-1}(c)$ is closed in $X$, is $f$ continuous? If the ...
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0answers
15 views

A class of sparse matrices whose inverse is also sparse?

In general the inverse of a sparse matrix is dense. A notable (but trivial) exception from that rule are diagonal matrices. Is there any other (broad) class of sparse matrices whose inverse is also ...
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1answer
22 views

Finding multiplicative inverse Euler's theroem

been struggling this whole day with trying to figure out the multiplicative inverse of 17 modulo 31 using Eulers theorem. We know that 31 is a prime, φ(n)=30, so i end up with 17^30=(cong)1 (mod 31). ...
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45 views

Is the linear operator $T(f)(x) = f(-x) + f(x)$ invertible?

My understanding is that the inverse of a linear operator will effectively "undo" the operation. Therefore to get the inverse of this operator we need to somehow subtract the f(-x). But I'm not ...
2
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1answer
31 views

Is the linear operator $T_f(x) = f'(x)$ invertible?

I think that $T_f(x)=f'(x)$ is invertible. This seems likely because it is a differential operator, and the inverse of a differential operator is the integral operator (though I'd like more ...
0
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1answer
19 views

Condition inverse $p$-adic number

Take $p$ prime, $n \in \mathbb{Z}_{>0}$ and $x \in \mathbb{Z}_p$. Suppose that $p$ isn't a divisor of $$x = (x_j + p^j \mathbb{Z})_{j \in \mathbb{Z}_{>0}},$$ then one can prove that the first ...
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7answers
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Why does this “miracle method” for matrix inversion work?

Recently, I answered this question about matrix invertibility using a solution technique I called a "miracle method." The question and answer are reproduced below: Problem: Let $A$ be a matrix ...
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5answers
62 views

Find inverse of 15 modulo 88.

Here the question: Find an inverse $a$ for $15$ modulo $88$ so that $0 \le a \le 87$; that is, find an integer $a \in \{0, 1, ..., 87\}$ so that $15a \equiv1$ (mod 88). Here is my attempt to answer: ...
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3answers
29 views

$A,B$ are matrices $3x3$ so that $B^2A=-2B^3$ and $B^3+AB^2=3I$ express $A^{-1}$ and $B^{-1}$ using $B$

I have the follow question : $A,B$ are matrices $3x3$ so that $B^2A=-2B^3$ and $B^3+AB^2=3I$ express $A^{-1}$ and $B^{-1}$ using $B$ I tried to "play" with the equations but I always get stuck with ...
0
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1answer
13 views

Understanding Multiplicative Inverse in RSA

Okay so I am reading up on RSA, trying to understand how it works, and I come across this $ x∈ℤp, x−1 ∈ℤp ⟺ \gcd(x,p) = 1$ Now it then gives an example, as follows: Lets work in the set $ℤ9$, ...
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17 views

Finding the $h'(x,y,z)$ if $h= p \circ q $ $p(x,y,z)=(x \sin y, x \cos y, z+y ), q(x,y,z)=(x^2,x+y,2e^z)$

I just want someone to check my work basically. Providing thoughts and insight, into possible mistakes: Finding the $$h'(x,y,z)$$ if $$h= p \circ q ,\ \ p(x,y,z)=(x \sin y, x \cos y, z+y ), \ \ ...
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0answers
11 views

Inverse of a product of real functions

Given $F(x) = L(x)G(x)$, with $L$ and $G$ real function strictly greater than zero. Suppose that F and G are decreasing functions (so that $F^{-1}$ and $G^{-1}$ exists). What can we say about the ...
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3answers
47 views

Why can't this mixed function be inverted?

Given the function $$y=Ax + B\sqrt x$$ where $A$ and $B$ are real constants, $x$ is real and $x > 0$ I want to find the inverse where $x$ is a function of $y$. ButI don't believe that's possible ...
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0answers
39 views

What is Lebesgue measure of sets of inverse prime numbers in $[0,1] $?

I would like to know if it is possible to know the lebesgue mesure of sets of inverse prime numbers in $[0,1]$ Note : I think should to know in the first if the sets of primes are infinit countable ...
2
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1answer
24 views

Block matrix pseudoinverse: symmetry of the inverse of a symmetric matrix

In the wiki page for block matrix pseudoinverses, there is a formula $$ \begin{pmatrix}A & B \\ C & D\end{pmatrix}^{-1}=\begin{pmatrix} (A-BD^{-1}C)^{-1} & ...
3
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2answers
61 views

Inverting a $3\times 3$ block matrix

Suppose that $a$ and $b$ below are scalars, $F$ a square matrix, $v$ a column vector. I'm trying to invert the matrix $M$ of the form $$ M=\begin{pmatrix} a & v' & 0\\ v & F & 0\\ 0 ...
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0answers
16 views

Finding the inverse of a square circulant matrix

I'm having difficulties of finding the inverse of the following square matrix, which consists of $2\times 2$ circulant matrices: $A = \left[\begin{matrix}x^{383} & x^{102} + x^{253} \\ ...
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2answers
26 views

Flip Values to get the opposite

Not sure of the name of what I need to do, but I used to do it all the time, and now i forget. I have values 1 - 10. I want 10 to become 1 and 1 to become 10. What is the formula to do this again? It ...
2
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1answer
22 views

Calculate the modular inverse of $2a$ given that of $a$

My problem is that I have to calculate some modular inverses of numbers that are related by multiplying by $2$, that is: Given $a$ and $x$ so that $ax\equiv1\mod n$ ($n$ being an odd number) I need ...
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3answers
77 views

Can we prove $BA=E$ from $AB=E$? [duplicate]

I was wondering if $AB=E$ ($E$ is identity) is enough to claim $A^{-1} = B$ or if we also need $BA=E$. All my textbooks define the inverse $B$ of $A$ such that $AB=BA=E$. But I can't see why $AB=E$ ...
0
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1answer
47 views

L2 norm of an inverse of a sum of matrices

I am trying to take the L2 norm of the following expression: $-(H^{-1} + bI)^{-1}v$, where $H$ is a psd matrix, b is a scalar, and $v$ is a vector. In particular I am having trouble with the first ...
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3answers
40 views

If a function has an inverse then it is bijective?

I have some trouble finding the answer to this, can someone help me out: If I have a general function $f$ with domain $X$ and codomain $Y$, I know nothing about the function (injective, surjective). ...
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Is there a polynomial $p$ such that it is bijective and $ p: \mathbb{Q}^n \rightarrow \mathbb{Q}$ for $ n>1$ ??

Let us define a polynomial $p$ defined as follow $$p: \mathbb{Q}^n \rightarrow \mathbb{Q}.$$ I acrossed this question in mind. Is there a polynomial $p$ such that it is bijective and $p: ...
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1answer
39 views

Inverse of the sum of the inverse of 2 non-invertible matrices

Given that the following square matrices are non-invertible: $\bf A$, $\bf B$, and (A+B) UPDATE: Assume $\bf (A+B)$ is invertible. and given that $\bf (A+I)$, $\bf (B+I)$, and $\bf ...
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2answers
34 views

Does injective imply each $x$ matches to a unique $y$?

Injective means one-to-one matching, as in each $y$ is matched by only one $x$. However, does this mean that each $x$ matches only to one $y$?
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1answer
38 views

Inverse Laplace

I want to calculate the inverse laplace of $$F(s)=e^{-3s}\frac{1+s}{s^3+2s^2+2s}$$ And i'm wondering if applying the derivative theorem is correct. To keep it simple it split them up: ...
0
votes
1answer
24 views

In what condition we have $(K^{-1})^\ast = (K^\ast)^{-1}$?

Suppose $X$ $Y$ are two finite dimensional Hilbert space. Assume $K$: $X\to Y$ is linear. My question is, in what condition of $K$ that $$(K^{-1})^\ast = (K^\ast)^{-1}?$$
4
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2answers
80 views

Is this matrix invertible?

I have been working on a proof and am stuck with showing that the below matrix is invertible. I am not interested in the explicit inverse, only showing it has a nonzero determinant as the existence of ...
2
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1answer
41 views

How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.

A while ago, I found that: $$f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt}.$$ I managed to prove it for a few functions, and I believe that it may be the case for all ...
0
votes
3answers
54 views

Inverse of partitioned matrices [closed]

A matrix of the form $$A=\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix}$$ is said to be block upper triangular. Assume that $A_{11}$ is $p \times p$, $A_{22}$ is $q \times q$ and ...
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119 views

Harder-Than-Seems Inverse of $f(x)=x^3-x-12$?

This may seem simple but I have had long days of frustration with finding the inverse of this: $$f(x)=x^3-x-12.$$ I got this on some homework and it did not ask for the inverse. However I wanted to ...
9
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3answers
849 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
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2answers
63 views

Clarification of definition of “inverse” with quaternions

From what I understand, the inverse of a matrix only exists if the matrix is square. I recently learned however that the inverse of a quaternion is the quaternion vector (1xn dimensions) where each ...
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2answers
29 views

Does for $T \in B(X)$ with $\|T\|>1$ exist $T^{-1}$?

Is it true if $\|T\|>1$, where $T \in B(X)$ for some Banach space $X$, then $T^{-1}$ exists? I suppose that for $\|T\|=1$ this isn't true? Because, if we suppose that inverse exists for such ...
0
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2answers
40 views

inverse a function with exponential and first degree polynom

I need some help to inverse this function: $$ y = a(e^{bx}-1) + cx + d $$ with $y(0)=d$ and $y(k)=0$ where $k$ is a constant. I don't know how to proceed. Thanks.
0
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1answer
57 views

Find the inverse function of $ f (x ) = x^2 - x - 2$

Find the inverse function of $ f (x ) = x^2 - x - 2$, where x is equal to or larger than 1/2. I tried to express it in form of $ (x - 1 )^2 = y + 2 $, but this is not true as the term in the middle ...
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0answers
23 views

Why does the inversion of this circle give a horizontal line y=i/2?

Inverting the circle centered at $(0,-i)$ with radius 1, gives the horizontal line $y = \frac{i}{2}$, but why does it have to be horizontal - Why not another straight line passing through the ...
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1answer
40 views

Inverse and composite functions [closed]

If $f(x)=\frac{x}{1-√x}$, $x≥0$ and $g(x)=3x+1, $ Solve the equation $(f^{-1}\circ g)(x)=9/16$. Hint:do not attempt to find $f^{-1}(x)$.
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3answers
49 views

Obtaining a Non-Singular Matrix from a Singular one by Perturbation

In a paper "http://www.math.cornell.edu/~nussbaum/papers/08-1.pdf" (page 264 Lemma 2) I encountered the following way of obtaining an invertible (non-singular) matrix from a non-invertible (singular) ...
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1answer
43 views

Understanding $F \circ \phi^{-1}$ in differential geometry

I am struggling with a question in elementary differential geometry. I thought I understood the basics until I read page 20 of The Geometry of Physics by T. Frankel. Suppose we have a manifold of ...
0
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2answers
30 views

Prove $xf(x) - \int_{0}^{x} f(t) \,dt = \int_{f(0)}^{f(x)} g(t) \,dt, g(x) = f^{-1}(x), \forall f(x)$?

I have proved it for all functions of the form $f(x) = x^{a}, \forall a, x,$ but I am not sure if it is true for all functions. Proposition $$xf(x) - \int_{0}^{x} f(t) \,dt = \int_{f(0)}^{f(x)} ...
2
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0answers
21 views

Closest line to point after non-linear map

I have a map on a vector space $M(\vec{r})$, defined as below. All components (vectors, matrices, everything) are reals in the unit range $[0,1]$. The map $M(\vec r)$ is defined as the sum of an ...