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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106 views

Calculating $\text{erf}^{-1}(z)$ for $z\in\mathbb{C}$

All the information I found about inverse error function $\text{erf}^{-1}(z)$ was about $z\in\mathbb{R}$. Also I found some Taylor expansions for it, but as the function is unbounded near $z=\pm1$, ...
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1answer
45 views

Injective and Surjective Functions on Sets

I'm fairly new to math proofs. I've been looking for some counterexamples to the following theorems, especially the second one. I haven't been able to think of a scenario. Are the following theorems ...
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1answer
49 views

Finding the Inverse Laplace transform using the Step and Shift theorems

I want to find the Inverse Laplace Transformation of the function given above. I used the step and shift theorems to come up with an answer. Can someone simply verify the answer. This is my first ...
10
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1answer
222 views

From matrices to bipartite graphs

Assume $G(A,B)$ is a bipartite graph and assume $L(G)$ is the adjacency matrix of its line graph. define $$B=[3\text{I}+L(G)]^{-1}$$. Is it always the case that for each edge $e=(a,b)\in G$, we have: ...
8
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1answer
340 views

Simple to state yet tricky question

Define $$A=\left[\mathrm I+\sum_{k=1}^{m_1}v_k v_k^T+\sum_{k=1}^{m_2}u_k u_k^T\right]^{-1},$$ where each $u_k$ and $v_k$ is a $0$-$1$ column vector, and for each $1\leq i \leq n$, the $i$th component ...
3
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2answers
1k views

A square matrix A is invertible if and only if det A ≠ 0. Use the theorem above to find all values of k for which A is invertible

$$\begin{pmatrix} k & k & 0 \\ k^2 & 25 & k^2 \\ 0 & k & k \end{pmatrix}?$$ I did a sample question before this one: $$\begin{pmatrix} k & k & 0 \\ k^2 & 16 & ...
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2answers
177 views

Tricky question in Matrices! [closed]

Define $$A=[\text{I}+\sum_{k=1}^{m}u_{k}u_{k}^T]^{-1}$$, where for each $u_k$ is a $0-1$ column vector. Prove that for every $1\leq k \leq m$ $$Au_{k}u_{k}^T\geq0$$ i.e. each entry of $Au_ku_k^T$ ...
0
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1answer
62 views

Laplace Transformation spring question

Here is the question: http://i.imgur.com/XAH2UnX.jpg I can't seem to get the answer. Are those values in the writing like 1N/m even relevant? Can someone give me some direction? Thanks!
2
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0answers
43 views

Ultrametric matrices and their inverse

A non-negative square matrix $A$ is ultrametric iff: $A(i,i)>\{A(i,k),A(k,i)\}\forall k,i$ $A(i,j)\geq\min\{A(i,k),A(k,j)\}\forall i,j,k$ It is well-known that the inverse of non-negative ...
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4answers
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Is $A + A^{-1}$ always invertible?

Let $A$ be an invertible matrix. Then is $A + A^{-1}$ invertible for any $A$? I have a hunch that it's false, but can't really find a way to prove it. If you give a counterexample, could you please ...
0
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1answer
60 views

Calculating Inverse function

Please help me with the following question: Calculate, if possible, the inverse of the following functions: (i) $f(x) = (2x - 2)^5$ (ii) $f(x) = (2x - 3)/4$ (iii) $f(x) = x^2 + 1,$ for $ x \geq ...
0
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0answers
53 views

Conversion of roots of a polynomial

I'm wondering, given a polynomial $P(x)$ with roots $r_i (1\le i\le n)$, how to determine the polynomial $Q(x)$ such that its roots are $r'_i=f(r_i)$. For example, if $P(x)=x^2-x-6=(x-3)(x+2)$ and ...
0
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4answers
209 views

If $g(x)=3+x+e^x$, then how do I find $g^{-1}(4)$?

If $g(x)=3+x+e^x$, then how do I find $g^{-1}(4)$? I took $g(x)=y$ and tried to solve the problem, but i could not get the solution.So, please help me by providing me the solution to my question.
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0answers
87 views

Can I find the Pseudoinverse (Moore-Penrose inverse) just by knowing the one-sided inverses of a matrix?

Consider a matrix such as $B = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \end{bmatrix}$. I know how to compute the right inverses (or in the case of $m\geq n$ the left inverses) and ...
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2answers
158 views

What is the inverse function of $f(x)=x/(1-x^2)$

Can you give me a hint for how the inverse function of $f\colon (-1,1)\to \mathbb{R}\colon f(x)=\frac{x}{1-x^2}$ looks? I need to show a homeomorphism!
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1answer
378 views

Trick: Substitution in inverse trigonometry.

My friends say, it is some what difficult to know, which trigonometric function has to be substituted in the inverse trigonometric equations, to get the correct solution. So, I thought to take up this ...
0
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1answer
21 views

Inverse of $\{a_1 A_1,…,a_n A_n\}$

$a_1,...,a_n\in \mathbb{R}$ $A_1,...,A_n$ are the rows of the invertible matrix A I am trying to find a regular formula for this. Is it possible? Thanks for help!
3
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0answers
757 views

Left inverse iff injective; right inverse iff surjective

For a function $f:A\to B$, the function $g:B\to A$ is called: a left inverse for $f$ if $g\circ f$ is the identity on $A$ (i.e., $g\circ f = {\rm id}_A$); and a right inverse for $f$ if ...
0
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1answer
65 views

Solution of matrix equations

$$A=\begin{bmatrix} 3 & -2 & -1 \\ 1 & 2 & 1 \\ -1 & 1 & 1 \end{bmatrix}, X= \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 1 \\ 7 \\ 2\end{bmatrix}$$ ...
1
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1answer
90 views

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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523 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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1answer
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 ...
0
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1answer
109 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} ...
0
votes
1answer
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 $(I+cH^{-1})^{-1}$ ...
0
votes
1answer
33 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: $$ ...
0
votes
2answers
222 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 ...
0
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0answers
37 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 ...
0
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1answer
45 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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1answer
148 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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277 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 ...
1
vote
1answer
61 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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253 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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227 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: ...
1
vote
1answer
62 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. ...
0
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1answer
69 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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vote
1answer
967 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 ...
0
votes
1answer
73 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 ...
1
vote
1answer
244 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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vote
1answer
228 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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1answer
39 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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88 views

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 ...
2
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3answers
50 views

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 ...
0
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1answer
34 views

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: ...
2
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0answers
44 views

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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1answer
123 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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0answers
47 views

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 ...
2
votes
1answer
189 views

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 ...
3
votes
7answers
277 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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vote
1answer
86 views

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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2answers
44 views

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 ...