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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Looking for a commutative ring satisfying certain conditions

I'm looking for a commutative ring $R$ (with unit) which is of characteristic 2 and which possesses elements $x$ and $y$ such that the following holds $x^2$ and $y^2$ are inverses of one another but ...
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1answer
150 views

Inverse of $f(x)=\sin(x)+x$

What is the inverse of $$f(x)=\sin(x)+x.$$ I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet. What about $$f(x)=\sin(a \cdot x)+x$$ where ...
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1answer
153 views

If matrix A is invertible, is it diagonalizable as well?

If a matrix A is invertible, then it is diagonalizable. Is it true or false?
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159 views

Solve equation $\tfrac 1x (e^x-1) = \alpha$

I have the equation $\tfrac 1x (e^x-1) = \alpha$ for an positive $\alpha \in \mathbb{R}^+$ which I want to solve for $x\in \mathbb R$ (most of all I am interested in the solution $x > 0$ for ...
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2answers
7k views

To invert a Matrix, Condition number should be less than what?

I see that there is a matlab tag in this site, so I ask my question here and not in stackoverflow although it is also related to programming in matlab. I am going to invert a positive definite matrix ...
4
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1answer
344 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 ...
4
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1answer
54 views

Invertible matrix of non-square matrix?

Is a matrix invertible only when it is a square matrix? What about a matrix of the order $m \cdot n$ with $m \gt n$ and such that it is row-equivalent to a row-reduced echelon matrix with more ...
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1answer
32 views

inverse of a point $p$ respect to the circle $|z-z_0 |= r$ in complex

I was solving a problem to find the inverse of a point $p$ respect to the circle $|z-z_0|=r$. In my question I had to find inverse of $1+i$ w.r.t circle $|z+1-2i| = 2$. I applied the formula $q = z_0 ...
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2answers
122 views

$\ln(x)$, $e^{x}$ and $\int \frac{1}{x}dx$ relationship

My math professor told me that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$ by the definition; so far so good. But how/why does $\ln(x)$ ($\int_1^x\frac{1}{t} dt$: by defintion) coincide with the inverse of ...
4
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1answer
209 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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2answers
79 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 ...
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127 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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1k views

Pseudo inverse of a product of two matrices with different rank

Let $V$ be an $n \times n$ symmetric, positive definite matrix (of rank $n$). Let $X$ be an $n \times p$ matrix of rank $p$. Define $A^- = (A^\top A)^{-1} A^\top$ as the pseudo inverse of $A$ when ...
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161 views

Find inverse function

Is it possible to get inverse of all be functions? For example, can we calculate inverse of $y=x^3+x$?
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1answer
677 views

How to invert a very regular banded Toeplitz matrix?

What's the best way to invert a simple Toeplitz matrix of the following form? $$ A = \begin{bmatrix} 1 & a & 0 & \ldots & \ldots & 0 \\\ a & 1 & a & \ddots & ...
4
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1answer
127 views

If $z$ is the unique element of a monoid such that $uzu=u$, is $u$ invertible?

This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success. Let $M$ be a monoid and $u\in M$. ...
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55 views

Inverse of an ordered pair?

Let $f: A \to B$ be a bijective function where $A = [0, 2\pi)$ and $B$ is the unit circle. Find the inverse of $f(\theta) = (\cos\theta, \sin\theta)$. I don't understand what it means to take the ...
4
votes
2answers
200 views

How to find the inverse cosine without a calculator

How to find the inverse of: $$\cos(c)=\frac{1}{3}$$ In other words, i'm trying to solve for c and without a calculator. If it's hard or not possible, then how would you go about solving inverses in ...
4
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2answers
140 views

Invertibility theorem on the boundary for a function between two closed 2D manifolds

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected, closed domain $D\subset\mathbb{R}^2$ including its boundary $\partial D$. I am interested in the local invertibility of $f$ ...
4
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117 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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6answers
87 views

Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$

If $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$. Here is what I have done so far. I have took $f'(x)=(1+x^2)^{1/2}$ and I have found $1/f'(0)$ which should equal $1$. I don't think this ...
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1answer
196 views

A formula for n-derivative of the inverse of a function?

Let $y=f^{-1}(x)$. As we know: \begin{align} \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{{f}'(y)} \end{align} Thereof we have: \begin{align} \frac{\mathrm{d^2} y}{\mathrm{d} ...
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118 views

Inverse of matrices with 3 parts!

I just wonder if there is any closed form solution for the inverse of matrices with following form, or if it's possible to decompose them. $ \left[\begin{array}{cccccccccc} {\color{red}1} & ...
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139 views

How do we know how many branches the inverse function of an elementary function has?

How do we know how many branches the inverse function of an elementary function has ? For instance Lambert W function. How do we know how many branches it has at e.g. $z=-0.5$ , $z=0$ , $z=0.5$ or ...
4
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1answer
55 views

Solving a set of non-linear matrix equations

Consider the following set of equations $$\begin{cases}PAQ^{-1}&=T \\ QBR^{-1}&=T\\ RCP^{-1}&=T, \end{cases} $$ where A,B,C and T are known real-valued $3\times3$ matrices and P, Q, R are ...
4
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1answer
58 views

Optimal series expansion for “invertability”

Motivation: Often when dealing with physical phenomena, deviations from the model must be considered, so a variable, say $x\in[0,1]$ will be replaced by a power series expansion: $$x'\ \to \ x(1+k ...
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2answers
72 views

Proving inverses with permutations?

Prove (if f and g are permutations) that $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$. My teacher gave me the hint that it has something to do with identity mapping, but that doesn't help me at all. ...
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1answer
96 views

solve $ y = (A+B^{-1})x $ for $x$

I wish to solve numerically for $x$, $$ y = (A+B^{-1})x $$ with $A, B$ positive definite. So, $$ x = (A+B^{-1})^{-1}y $$ I would like to avoid calculating $B^{-1}$ since that's generally bad. ...
4
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86 views

inverse of a tridiagonal matrix

Let $${A_{n \times n}} = \left[ {\begin{array}{*{20}{c}} {-2}&{1}&{}&{}&{}\\ {1}&{-2}&{1}&{}&{}\\ {}&{1}&{\ddots}&{\ddots}&{}\\ ...
4
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1answer
194 views

Inverse Function Differential Equation [duplicate]

For the differential equation $$\frac{d}{dx}[y(x)]=y^{(-1)}(x)$$ where $y^{(-1)}(x)$ is the inverse of $y(x)$, find y(x). I gave up on finding the solution analytically pretty quickly and decided ...
4
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1answer
144 views

Inverse function of $y=x+kx^3$

I want to invert the following function with respect to $x$: $$f(x, k)=x+k x^3$$ where typical values for $x$ are between $0$ and $100$ and typical values for $k$ are between $-0.00005$ and ...
4
votes
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216 views

nonegative inverse eigenvalue problem

I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form: $$ \mathbf{M} = \begin{vmatrix} \mathbf{A} & \mathbf{b} \\ ...
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votes
2answers
80 views

norm of inverse less than 1

I just wanna ask if what I am doing here make sense: Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
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0answers
79 views

What function satisfy: $f(x)+f^{-1}(x)=2x$?

What function satisfy: $f(x)+f^{-1}(x)=2x$? I have tried to substitute $x=f(x)$ to get $f^{(2)}(x)+1=2f(x)$ and subsequently plug in values to try to find $f(x)$ but to no avail. Please help thank ...
4
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0answers
162 views

Inverse of two matrices multiplied [closed]

I've been asked to find the inverse of $AB$ where $A$ and $B$ are: $$A=\begin{bmatrix}5 & 3 \\4 & 2\end{bmatrix}$$ $$B=\begin{bmatrix}2 & -3 \\1 & 3\end{bmatrix}$$ My answer: What I ...
4
votes
1answer
139 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
4
votes
0answers
50 views

Is there always a smooth variant of a homoeomorphism between smooth manifolds?

Let $M$ and $N$ be smooth homeomorphic manifolds. Let $h:M\rightarrow N$ a homeomorphism. Does there exist $r:M\rightarrow N$ that is still a homeomorphism and additionaly smooth? Can it be chosen ...
4
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0answers
78 views

Is the inverse of any elementary function asymptotic to some elementary function?

Is the functional inverse of any elementary function asymptotic to some elementary function ? For instance Lambert's $W(z)$ is asymptotic to $ln(z)$. See ...
4
votes
0answers
83 views

Explicit quasi-inverse of Künneth-isomorphism?

With $A_X$ the complex of $\mathbb{R}$-differential forms on $X$, the Künneth theorem states that \begin{align*} A_X \otimes A_Y &\to A_{X \times Y}, \\ (\omega,\eta) &\mapsto {\rm ...
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votes
7answers
671 views

How come the function and the inverse of the function are the same?

What is the inverse of the function: $$f(x)=\frac{x+2}{5x-1}$$ ? Answer: $$f^{-1}(x)=\frac{x+2}{5x-1}$$ Can one of you explain how the inverse is the same exact thing as the original equation?
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7answers
284 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 ...
3
votes
3answers
3k views

How do I find the inverse function of a polynomial with $x^5$?

I've been stumped on this problem for hours and cannot figure out how to do it from tons of tutorials. Please note: This is an intro to calculus, so we haven't learned derivatives or anything too ...
3
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4answers
95 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 ...
3
votes
3answers
185 views

Is $\sqrt{x^2}$ always $\pm x?$

I am wondering if this holds in every single case: $$\sqrt{x^2} = \pm x$$ Specifically in this case: $$\sqrt{\left(\frac{1}{4}\right)^2}$$ In this one we know that the number is positive before ...
3
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7answers
167 views

$f \circ g =\operatorname{ id}$ and $g \circ f \neq \operatorname{id}$?

Are there two functions $f$ and $g$ s.t. $$f \circ g = \operatorname{id}$$ but $$g \circ f \neq \operatorname{id}?$$ Could someone give an example or a proof that this is impossible? This must be ...
3
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2answers
485 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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8answers
206 views

How do I solve $x^5 +x^3+x = y$ for $x$?

I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$? The explicit question out of the ...
3
votes
2answers
83 views

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 $\Theta(log_2 n)$ ?
3
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2answers
94 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
votes
4answers
128 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.