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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Solution of matrix equations

$$A=\begin{bmatrix} 3 & -2 & -1 \\ 1 & 2 & 1 \\ -1 & 1 & 1 \end{bmatrix}, X= \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 1 \\ 7 \\ 2\end{bmatrix}$$ ...
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Finding the inverse function

The question is to find the inverse function of $$f(x)=x-(2\sqrt{x})+1$$ I first found that the domain of definition is $\,x\ge 0$ Then studied the variation of the function and it is decreasing ...
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387 views

Showing a function is bijective and finding its inverse

The function f: ℝ2-> ℝ2 is defined by f(x,y)=(2x+3y,x+2y). Show that f is bijective and find its inverse. I've got so far: Bijective = 1-1 and onto. 1-1 if (2x1+3y1,x1+2y1)=(2x2+3y2,x2+2y2) Then ...
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70 views

derivative of product of 2 inverse matrices

I was trying to differentiate the equation below: $$ \frac{\partial a^T X^{-T}X^{-1}a} {\partial X} $$ where X is invertible but not symmetric and $X^{-T}$ means transpose of inverse of X. In the ...
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103 views

Find the inverse z-transform of $E(z)=\frac{z+1}{(z-1)(z-0.6)}$

$$\begin{align} E(z)&=\frac{z+1}{(z-1)(z-0.6)}\\ \frac{z+1}{(z-1)(z-0.6)}&=\frac{A}{(z-1)}+\frac{B}{(z-0.6)}\\ z+1&=A(z-0.6)+B(z-1) \end{align}$$ set z=0.6: $$\begin{align} ...
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32 views

About the inverse matrix of the form $(I+cH^{-1})^{-1}$.

Given $(I+cH^{-1})^{-1}$, where $c$ is a constant and $H$ is a $\mathbb{R}^{n\times n}$ matrix. Suppose $(I+cH^{-1})^{-1}$ has a inverse matrix. Is there any way to calculate $(I+cH^{-1})^{-1}$ ...
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32 views

Inverse Transformation

Consider the coordinate transformation $$ \varphi\colon\mathbb{R}^2\to\mathbb{R}^2, (x,y)\mapsto (y-\arctan(x),y+\arctan(x)). $$ To make it more easy, I set: $$ ...
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178 views

Inverse functions and tangent line

Let $f(x) = \frac14x^3 + 12x + 6$ and let $y = f^{-1}(x)$ be the inverse function of $f$. Determine the $x$-coordinates of the two points on the graph of the inverse function where the tangent line is ...
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34 views

Determinant after rank 1 update of a singular matrix

The rank-1 update to the inverse of a matrix and rank-1 update to the determinant of a matrix are closely related. I would like to compute the determinant of a rank-1 updated singular (rank-1 ...
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41 views

differentiate of an inverse function of mixed exponential and algebraic form

Let $f(x)= e^{2x} + x^5 + 1$ Find $(f^{-1})'(2)$ Find $(f^{-1})''(2)$ There is a missing link in my brain with regards to dealing with a function containing exponential and algebra. :/ I'm ...
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138 views

How can we derive pseudo inverse of a matrix from its Singular value decomposition?

For a matrix $M$ with its singular value decomposition $UΣV^T$, the pseudo inverse of $M$, i.e., $M^+$ is $VΣ^+U^T$. How can I derive the pseudo inverse(Moore–Penrose) $M^+$ from the singular value ...
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233 views

Multiplicative Inverses in Non-Commutative Rings

My abstract book defines inverses (units) as solutions to the equation $ax=1$ then stipulates in the definition that $xa=1=ax$, even in non-commutative rings. But I'm having trouble understanding why ...
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54 views

Composition of function with it's inverse on subdomains

I have a short question. We have to check the following statements and tell for which one the equal sign holds. Let $M \subset \mbox{domain } f$ and $N \subset \mbox{Im } f$. ...
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213 views

Calculus inverse tangent line

Question is: Let $ f(x)=(1/2)x^3+6x+4 $ and let $y=f^{-1}(x)$ be the inverse function of f. Determine the x-coordinates of the two points on the graph of the inverse function where the tangent line ...
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182 views

Operation counts for algorithm using Gaussian elimination to find A^(-1)

I need help determining the operation counts of my algorithm that uses Gaussian elimination to find the inverse of a matrix. Can anyone help me? Here is my algorithm: ...
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1answer
57 views

3-D function that follows an inverse square law, but has an overall integral equal to a constant

I'm currently trying to figure out a 3D function which follows the "inverse square law" along any given ray drawn from 0,0,0 coordinates, but whose -inf..inf integral over all arguments converges. ...
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63 views

Delta function that obeys inverse square law outside its (-1; 1) range and has no 1/0 infinity

Does anybody know if such function exists? As I understand it, the function $$\frac{1}{x^2}$$ itself could be used as a delta function if it had no 1/0 infinity. That is why I'm in a search of an ...
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781 views

Solving System of Congruence using Chinese Remainder Theorem

I'm trying to solve a system of congruence using CRT $$x≡2\pmod3\\ x≡3\pmod5\\ x≡2\pmod7$$ My approach is First calcuating $m_1,m_2,m_3$ then M followed by inverses of $m_1,m_2$ and $m_3$ and ...
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72 views

A question about similar matrices: $Id$ and $-Id$

Currently, I'm trying to understand the idea of matrix similarity. As a toy example, I am thinking about $Id$ and $-Id$. Now, I do not think that these matrices are similar, and here is my proposed ...
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198 views

Finding the inverse of the sum of two symmetric matrices A+B

Consider calculating the inverse of matrix sum $$A+B$$ where A is a symmetric dense matrix while B is a symmetric block diagonal matrix. I am interested in finding an efficient approach to update ...
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188 views

Inverse Laplace Transform for $F(s) = (9s-24)/(s^2-6s+13)$

Find the inverse Laplace transform of $\displaystyle F(s) = \frac{9s-24}{s^2-6s+13}$. I have tried factoring out a $3$ from the top and putting it into the form of $\displaystyle\frac{b}{(s-a)^2+b^2}$ ...
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37 views

Updating eigen decomposition for a matrix after some row changes

Let us say we have a matrix $A$ which has eigen decomposition $$A=UDU^{-1}$$ If some of the rows of A are changed by multiplying a constant positive value, is there a simple way to update the eigen ...
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Question about the Miller Theorem on inverse of sum of two matrices.

The following is a well known theorem on the inverse of $(A+B)$. (Link to the paper: http://www.jstor.org/stable/2690437) Theorem. Let $A$ and $A+B$ be nonsingular matrices, and let $B$ have rank ...
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Inverting a quartic equation of state

I have the following equation (which is an adaptation of the Beattie-Bridgeman Equation of State): $$ P = \frac{RT}{V} + \frac{B}{V^2} + \frac{C}{V^3} + \frac{D}{V^4} $$ This is a function of the ...
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Inverse function without the original function

I am going through this paper, 'Certifiable Quantum Dice Or, True Random Number Generation Secure Against Quantum Adversaries' by Vazirani and Vidick. In 'Our results' section on the page 2, it says: ...
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Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
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102 views

Invertible functions and their properties

If an n × n matrix A is singular, then the columns of A must be linearly independent. Is this true? Invertible functions must be bijective Invertible functions must have square matrices Invertible ...
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Finding the number of the real roots of $a^x=g(x)$ where $g(x)$ is the inverse function of $f(x)=a^x$

Question : Let $a$ be a constant which satisfies $0\lt a\lt 1$. Letting $g(x)$ be the inverse function of $f(x)=a^x$, then find the number $N$ of the real roots of $f(x)=g(x)$. Motivation : This is ...
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Inverse function of $x\mapsto \sqrt[x]x$ on $\left[0,e^{-1}\right]$

Why is it, that the inverse of $\sqrt[x]x$ is given by the infinite power tower in $x\in[\frac1e;e]$, but not in $x\in[0;\frac1e]$? I know that the power tower diverges on that interval, but even if ...
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274 views

How to find the $f^{-1}(x)$ of $f(x)=x^{3}-12x+\frac{48}{x}-\frac{64}{x^{3}}$

It is a question from a quiz. The following is the whole question. Let \begin{eqnarray} \\f(x)=x^{3}-12x+\frac{48}{x}-\frac{64}{x^{3}} , \space x\in (-\infty ,0), \end{eqnarray} find ...
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Determining if a homomorphism is an isomorphism

Let $T \in \mathcal{L}(V)$, where $\mathcal{L}(V)$ is the set of linear operators mapping a vector space $V$ to itself, and let $U$ be an isomorphism from $V$ to another vector space $W$. We claim ...
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Uniqueness of Inverses in Groups Implies Associativity Holds?

I was checking multiplication tables for groups with 4 elements, to see which tables "passed" the group axioms of closure, associativity, identity and inverses. But then I had a question, so hopefully ...
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Inverse of an equation with $X_i$ and $X_{i -1}$

I'm trying to figure out the Inverse Transform Method for stats and in my profs. slides there is this: F(x) = cumulative dist. function, F(x) = Pr(X<=x) For the $i$th segment $(i = ...
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Computing an inverse modulo $25$

Supposed we wish to compute $11^{-1}$ mod $25$. Using the extended Euclid algorithm, we find that $15 \cdot 25 - 34 \cdot 11 =1$. Reducing both sides modulo $25$, we have $-34 \cdot 11 \equiv 1$ mod ...
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computing the expection of the inverse matrix

Is it possible to compute the following expectation in closed form? $E_{\alpha,\beta}\{(\exp(\alpha) A_1+\exp(\beta) A_2 + A_3)^{-1}\} $ where $\alpha$ and $\beta$ are Gaussian distributed with mean ...
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275 views

Simplifying an inverse trig function?

I am trying to figure out how to simplify this expression but I am not quite sure how these inverses work. What sort of approach should I take for this equation? ...
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3answers
86 views

How to get the inverse of this function?

I have the function f(x) = (1+8x) / (3-3x). I have been stuck on trying to get the inverse of this function by isolating for y. I ended up with: x = (1+8y) / (3-3y) but I am not quite sure where to ...
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30 views

Inversing a function

I'm having some problems calculating the inverse of this function: $f(u,v)=(u+v,v-u^2)$, its domain is $D=\{(u,v)$ in $\Bbb R^2 : u>0\}$ Thanks in advance.
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1answer
60 views

Left inverse of function $f:\mathbb R \to \mathbb R$

Let there be a function $f: \mathbb R \to \mathbb R$ given by $$f(x)=\begin{cases}5x + 2&x\geq 1\\x-1&x<1 \end{cases}$$ Give an example of a Left inverse of $f$, and prove that it is ...
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69 views

Finding inverse of functions[methods of]

I am now trying to understand functions, inverses and composites. I must admit am not getting a thing. But following some leads, I managed to work one as below. Is this a good understanding on hows ...
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What are the methods to find inverse of a function?

What are the different methods to find inverse of a function? The one that I have been taught in high school is by converting $y=f(x)$ into $x=y(x)$. But, this requires a lot of simplification and may ...
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What is the general area of mathematics to which this example belongs?

In elementary college-level calculus courses, I've given students a problem which reduces to this: Given $f(p,q)$ and a relation $p=g(q)$ use substitution to derive $\mathfrak{f}(p)$ then proceed ...
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57 views

Find x in polynomial given value of inverse

I'm studying for a test and this question has me really stumped: $f(x) = 2x^3+5x+3$. Find x if $f^{-1}(x) = 1$ I don't know how I am supposed to figure out the inverse of this polynomial. I used ...
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Is the inverse operation on matrices distributive?

For example, is the following true: $$(A + B)^{-1} = A^{-1} + B^{-1}$$ If $\det(A) \ne 0$, $\det(B) \ne 0$, and $\det(A + B) \ne 0$.
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208 views

Inverse Laplace Transform via residues

I have $\frac{1}{2 \pi i} \int_{\infty-iT}^{\infty+iT} \frac{e^{-s(1-t)}-e^{st}}{-s+e^{-s}-1} ds$ and I am trying to solve it using a contour. So I could have t>0, or t<0. I have a pole at 0. For ...
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Calculate the inverse for $\arctan(x^2+1), x≥0$

I have no idea how to solve this problem. Calculate the inverse for the function: $$f(x) = \arctan(x^2+1),\quad x≥0 .$$ Also specify $D_{f^{-1}}$ and $V_{f^{-1}}$. I would really ...
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137 views

Definite integral - Please point to me my mistake

This emerged while I was investigating this question, i.e. the solution to the definite integral $$I_x = \int_0^\infty\left(5x^5+x\right)\operatorname{erfc}\left(x^5+x\right)\,dx$$ In a comment, its ...
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1answer
58 views

Sharp bound on off-diagonal entries of matrix with 1's on diagonal to make matrix invertible

Suppose $A$ is an $n \times n$ matrix with all 1 on the diagonal. What is the sharp bound $\epsilon(n)$ so that $A$ is invertible if all off-diagonal entries of $A$ have absolute value less than ...
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Inverses of Modulo N

It's easy to show that relatively prime numbers have inverse mod n via the Euclidian Algorithm-How do you show that they don't necessarily have an inverse if they aren't relatively prime? I would ...
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372 views

Find the Inverse function of f. $f(x)=1+\sqrt{1+x}$

I found the Inverse of the function, $f^{-1}(x)= x^2-2x$. The back of my pre-cal book gives me the inverse of the function and the domain. What I don't understand is, how the domain comes to be $x ...