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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Why is $3^n$ not in $\Theta(2^n)$

How is it that $3^n$ not in $\Theta(2^n)$, while $log_3 n$ is in $\Theta(log_2 n)$ ?
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2answers
89 views

Does $\exp(\ln(I+A))=I+A$ when $\|A\|<1$?

For matrices, I know certain equalities like $e^{A+B}=e^Ae^B$ aren't always true. I'm curious, do $\exp$ and $\ln$ serve as inverses? I saw earlier that if $\|A\|<1$, then $\ln(I+A)$ converges. My ...
3
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2answers
471 views

What does $M^{-1}RM$ represent?

I'm a bit confused about the use of $M^{-1}RM$ where $R$ is a transformation matrix. Actually I was looking at the script here which reads and renders bvh files. But, I could not understand the ...
3
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4answers
295 views

How do you take the multiplicative inverse of a p-adic number?

I am reading the wiki page for p-adic numbers and it states that they are a field extension of the rationals so each member has to have a modular multiplicative inverse. So how would I take the ...
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3answers
296 views

How to invert this function? (Inverse exponential function with arctan)

How to invert this function? $$ y = e^{\arctan(x^5)} $$
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165 views

Find the inverse of a $4\times4$ matrix

My matrix looks like this: $$\left(\begin{array}{rrrr} 1& 1 & 1 & 1\\ 1& -1 & 1 & 0\\ 1& 1 & 0 & 0\\ 1& 0 & 0 & 0 ...
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4answers
56 views

Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$

I have a function: $$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$ With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...
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2answers
51 views

Working with groups. Finding the inverse of some $S_9$

I want to compute the inverse of: $\begin{pmatrix} 1&2&3&4&5&6&7&8&9\\3&2&1&6&5&9&4&8&7 \end{pmatrix}$ Sorry about alignment(they are ...
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4answers
120 views

Inverse of $(e^x - e^{-x})/2$

What is the inverse of the function $f(x)=\frac{e^x - e^{-x}}2$? I tried replacing $e^x$ by a variable but I still can't get it.
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4answers
188 views

Is there a good intuitive way to understand why matrix B is inverse of A when matrix A|I is turned into I|B

I'm looking for some help with my intuition of basic matrix operations, specifically finding a matrix's inverse (as per my subject line). I have no problems with the steps. The basic row operations ...
3
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2answers
102 views

Nilpotent matrices and inverses

Can somebody give me a hint for showing that: The matrix $A+I$ is invertible if there is an integer $k\gt 0$ so that $A^k=0$.
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The relation between an exponential function and a logarithmic function

I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're ...
3
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2answers
289 views

Finding inverse of a $3\times 4$ or $4\times 3$ matrix

Now I have no problem getting an inverse of a square matrix where you just calculate the matrix of minors, then apply matrix of co-factors and then transpose that and what you get you multiply by the ...
3
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2answers
57 views

If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$

Consider the statement: If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$. My book tells me this is suppose to be false, but I don't understand why. We know: If $f:X\to Y$ has ...
3
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3answers
178 views

Determine the greatest interval where the function is invertible

The assingment is to determine the greatest interval around $x=0$ where the function: $$f(x)=x^5-5x+3$$ is invertible. After that, determine $(f^{-1})'(3)$ I have totally forgotten all about ...
3
votes
1answer
47 views

The rank of general inverse of $A$ times $A$?

Supposing $X$ is the general inverse of $A$, that $AXA = A$. Then $XA$ is idempotent, that is $(XA)(XA) = XA$. Why is the rank of $XA$ equal to the rank of $A$ ? Thanks.
3
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71 views

Given the product of a unitary matrix and an orthogonal matrix, can it be easily inverted _without_ knowing these factors?

Given the product $M$ of a unitary matrix $U$ (i.e. $U^\dagger U=1$) and an orthogonal matrix $O$ (i.e. $O^TO=1$), can it be easily inverted without knowing $U$ and $O$? Sure enough, if $M=UO$, then ...
3
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4answers
138 views

How to show $AB^{-1}A=A$

Let $$A^{n \times n}=\begin{pmatrix} a & b &b & \dots & b \\ b & a &b & \dots & b \\ b & b & a & \dots & b \\ \vdots & \vdots & \vdots & ...
3
votes
1answer
93 views

Find whether or not an inverse exists algebraically

Is there an algebraic(without graphs) way to determine the existence of a function's inverse without using calculus? I'm an undergrad engineer and can obviously solve this using basic calculus, but ...
3
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3answers
165 views

Proof of Matrix Norm (Inverse Matrix)

Show for any induced matrix norm and nonsingular matrix A that $$ \left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1} $$ where $$ \left\|A^{-1}\right\| = ...
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99 views

What is the inverse of $f(x)=\frac{e^x+e^{-x}}{e^x-e^{-x}}$?

please help me to find out the inverse this function, $$f(x)=\frac{e^x+e^{-x}}{e^x-e^{-x}}$$ I know that, let $$y=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$$ and if I find $x=\cdots$ then that is the ...
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3answers
130 views

Finding inverse of a matrix

This question is in my assignment. We are not allowed to use any symbol to represent any elementary row and column operations used in the solution. We must solve it step-by-step. Please help me to ...
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2answers
93 views

A bijective mapping from $\mathbb N^k$ to $\mathbb N$?

Having $k$ numbers $N_i\in\mathbb{N}$, I'm looking for a bijective mapping $f:\mathbb{N}\times\ldots\times\mathbb{N}\rightarrow\mathbb{N}$ So that ...
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159 views

Proof with functions and inverse - Spivak

How does he know that $f^{-1}$ is one-one? Doesn't he have to prove that? Or is he applying his first theorem in the chapter to $f$? That is $f$ is a function if and only if $f^{-1}$ is ...
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1answer
51 views

If $A$ is an $n \times n$ matrix such that $A^3 = O_{3}$, show that $I - A$ is invertible with inverse $I + A + A^2$

So this question is basically a proof. If $A$ is an $n \times n$ matrix (so square) which satisfies the condition $A^3 = O_{3}$ ($A^{3}$ gives the $3 \times 3$ zero matrix), then show that $(I - A)$ ...
3
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1answer
210 views

Invertible Derivative

I'm trying to brush up on some differential geometry, but there's a subtle point I don't understand. Suppose $h$ is a diffeomorphism. Then the lecture notes here suggest that it's derivative $df_x$ is ...
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2answers
121 views

Formula for Nth Derivative of Matrix Inverse

I was looking for an equation for the nth derivative of a matrix inverse, ie $\frac{d^n \bf{A}^{-1}}{dx^n}$ I know that the first derivative $\frac{\text{d} \bf{A}^{-1}}{\text{d}x} = -\bf{A}^{-1} ...
3
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58 views

Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?

Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how? This seems like an easy question, but ...
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1answer
104 views

Inverse of a polynomial function

I want to find the inverse of $f(x)=\frac{3}{4}x^2-\frac{1}{4}x^3 $ when $0<x<2$. According to wolfram the answer is inverse I would like to know how can I find wolfram's inverse.
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53 views

Linear Algebra determinant and rank relation

True or False? If the determinant of a $4 \times 4$ matrix $A$ is $4$ then its rank must be $4$. Is it false or true? My guess is true, because the matrix $A$ is invertible. But there is ...
3
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1answer
100 views

does invertibility of product imply invertibility of each term of product?

Suppose $\mathcal{H}$ is a Hilbert space and the product $T_1T_2 \in B(\mathcal{H})$ is invertible. Does this imply that both $T_1, T_2$ are invertible ? I am unable to prove this since, unlike the ...
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1answer
546 views

Matrix Pseudo-Inverse using LU Decomposition?

What is the step by step numerical approach to calculate the pseudo-inverse of a matrix with M rows and N columns, using LU decomposition? So far, I have found this, but it uses singular value ...
3
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1answer
196 views

Dense pre-images implies continuous right inverse?

Suppose $f : \mathbb R \to \mathbb R$ is such that pre-image of every point under $f$ is dense in $\mathbb R$. This, of course, implies that $f$ is surjective, and hence has a right inverse ...
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222 views

Commutative monoid, unsure of how to deal with negative elements, inverses and subtraction

We are working with a commutative monoid. Subtraction might be useful for us. However, we're not sure how to proceed -- negative elements have no meaning. How do we deal with allowing subtraction ...
3
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1answer
2k views

Inverse function of a polynomial and its derivative

I know it is a simple problem but I am having trouble. Here is what I have so far: Let $f(x) = x^5 + 2x^3 + x - 1$ a) Find $f(1)$ and $f'(1)$ I have a) done. $f(1)$ is $3$ and $f'(1)$ is ...
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35 views

PID question in Ireland and Rosen

Context: In Ireland and Rosen's 'A classic introduction to number theory' on page 11, the proof that in a PID$=R$, there is an integer $n$ such that, for a prime $p$ and any $b\in R$, $p^n \mid b , ...
3
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2answers
994 views

Calculating Moore-Penrose pseudo inverse

I have a problem with a project requiring me to calculate the Moore-Penrose pseudo inverse. I've also posted about this on StackOverflow, where you can see my progress. From what I understand from ...
3
votes
1answer
285 views

Inverse of $(A + B)$ and $(A + BCD)$?

Consider $A$ as an arbitrary matrix and $B$ as a symmetric matrix. Since $B$ is symmetric, therefore, it can be written as a $\Gamma \Delta \Gamma'$, where $\Delta$ is a diagonal matrix with ...
3
votes
1answer
71 views

Why $ g(p) = 0.5 p^{-0.2} + 0.5 p^{-0.5} $ has a well-defined inverse that is continuous and strictly decreasing.

A book that I am reading claims the following about the function $ g(p) = 0.5 p^{-0.2} + 0.5 p^{-0.5} $ (which is a demand function): Formal arguments based on the Intermediate Value Theorem and ...
3
votes
1answer
175 views

Need help finding inverse under $a\circ b = a^b b^a$

I'm going through some problems in Theorems, Corollaries, Lemmas, and Methods of Proof, and I'm stuck at a certain problem that seemed very interesting until I couldn't solve it for the life of me. ...
3
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2answers
76 views

How adjacency matrix shows that the graph have no cycles?

Let $G$ a directed graph and $A$ the corresponding adjacency matrix. Let denote the identity matrix with $I$. I've read in a wikipedia article, that the following statement is true. Statement. $I-A$ ...
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78 views

Bijection, and finding the inverse function

I am new to discrete mathematics, and this was one of the question that the prof gave out. I am bit lost in this, since I never encountered discrete mathematics before. What do I need to do to prove ...
3
votes
1answer
29 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
56 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} ...
3
votes
1answer
85 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 ...
3
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1answer
39 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
3
votes
1answer
118 views

Contradiction in inverse Laplace transform problem with Mellin's inverse formula?

Let say we have to solve a given differential equation $$ty''+y'+ty=0$$ $$y(0)=1,\ y'(0)=0$$ (which is Bessel equation with the solution $y=J_0 (t)$, of course) with the Laplace transform. Then we ...
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2answers
38 views

Calculate the product ST, and infer from it the inverse of T.

S=\begin{pmatrix} 1/2 & 1/2 & 0\\ 1 & 0 & 0\\ -3/2 & 0 & 1/2 \end{pmatrix} T= \begin{pmatrix} 0 & 1 & 0\\ 2 & -1 & 4\\ 0 & 3 & 2 \end{pmatrix} I ...
3
votes
1answer
64 views

Inverse of Cartan matrix

The Cartan matrix of the root system $A_n$ looks like, denote it by $A'_n$ $$A'_n= \begin{bmatrix} 2 & -1 & 0 & 0&\ldots & 0 \\[0.3em] -1 & 2 & -1 ...
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2answers
237 views

Conditions for a matrix to be invertible

Let $n \geq m$ and let $C$ be a $n \times m$ full rank matrix, that is $rank(C) =m$. Considering that $D$ is a diagonal positive semidefinite matrix, under which conditions is the $ m \times m$ matrix ...