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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Jacobian Method for inverse kinematics

I have big problem. I have to solve inverse kinematics for a manipulator with 6-DOF using jacobian method. From what I know to do that I need to have matrix of transformation and Denavit–Hartenberg ...
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88 views

The Matrix Inversion Lemma: the General Case

I find it is hard to understand the application senario of the Matrix Inversion Lemma in non-special cases. Suppose I already computed $A^{-1}$ and want to find $\left(A+UCV \right)^{-1}$. The Matrix ...
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23 views

function inversion and the horizontal shift

I am currently doing inverse functions and graphing radical equations of the form $y=a\sqrt{x-h}+k$ with my algebra class and one of my students asked me the following question. "Why is it that we ...
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31 views

The Group of Complex Continuous Functions?

Let $C(\mathbb{C},\mathbb{C})=\{f:\mathbb{C} \rightarrow \mathbb{C}\,|\,f $continuous $\}$ be the set of all continuous functions from the complex plane to itself and consider the composition ...
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61 views

arrow structure matrices and Sherman-Morrison-Woodbury

I have two questions regarding "arrow structured" matrices and I'll be grateful if you can give more insights about them: 1- If A is an n-by-n SPD and has the arrow structure, e.g. A=[x x x x;x x 0 ...
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31 views

bound on matrix inverse with different elements

I'm hoping that someone can point me to some literature on the following. Is there a way to bound the inverse of a matrix if I change the value of 1 element in that matrix. Let's say I have a matrix ...
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55 views

Sherman-Morrison formula for rank 1 update

If $A$ is nonsingular and if for a particular $i$ and $j$ there is no way to make $A$ singular by changing $a_{ij}$ (rank-$1$-update), then using the Sherman-Morrison formula, what can we conclude ...
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1answer
69 views

Express parametric curve as graph of a function

I have a parametric curve in $\mathbb{R}^2$ given by $$ t\mapsto f(t)\left(\begin{array}{c}1\\1\end{array}\right)+\sqrt{-f'(t)}\left(\begin{array}{c}1\\-1\end{array}\right),\quad ...
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1answer
168 views

How to reverse modulo of a multiplication?

I am primarily a programmer (rather than a mathematician) and have recently come across a coding problem where I must invert a function which is the the modulo of a multiplication (given certain ...
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2answers
194 views

Taylor series of the inverse of $x^4+x$

I would like to expand the inverse function of $$g(x) := x^4+x $$ in a taylor series at the point x = 0. I calculated the first and second derivate at x = 0 with the rule of the derivation of an ...
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0answers
64 views

Is the pseudoinverse of a singular, lower triangular matrix itself lower triangular?

Suppose $L\in\mathbb{R}^{n\times n}$ is a singular, lower triangular matrix. Is its psuedoinverse, $L^\dagger\in\mathbb{n\times n}$, also lower triangular? I have already proved by induction that the ...
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1answer
66 views

Bromwich integral of $1/s^k$ with k real (non integer) and $1<k$

Is there a simple way to compute the inverse laplace transform of $1/s^k$ with k non integer using Bromwich integral (basically without using the known laplace transform of $t^n$)?
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44 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 ...
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4answers
96 views

What is the proper way to find the inverse of a function?

I am a little confused on the subject of inverse functions and the methods used to do the transformation from function to inverse. How do you make an inverse? Just so i can avoid any ambiguity in my ...
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1answer
115 views

Implicit Function Theorem to show no function can be one to one

Apply Implicit Function Theorem to show that no $C^1$ function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ can be one to one near any point of its domain. Repeat the proof by using Inverse Mapping Theorem ...
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1answer
35 views

Find if the system $(x(t-1))^2 + x(t) +(x(t+1))^2 = y(t)$ is invertible

If there wasn't the $x(t)$ term, I could use $x(t) = x$ and $x(t) = -x$ to disprove invertibility, but I can't think of two functions that give the same $y(t)$ in this case. When I tried proving ...
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96 views

Apply Implicit Function Theorem

Apply Implicit Function Theorem to show that no $C^1$ function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ can be one to one near any point of its domain. Repeat the proof by using Inverse Mapping Theorem ...
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32 views

Proves of identities in inverse trigonometry

Can someone please help me prove the following results from inverse trigonometry? $$\tan^{-1}x + \tan^{-1}y = \pi + \tan^{-1}\frac{x+y}{1-xy}( x>0, y>0, xy>1)$$ and $$\tan^{-1}x + ...
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39 views

Finding the inverse of a function involving |x|

I need to find the inverse of the following function $ f:(-1,1) \rightarrow \mathbb{R} $ $ f(x) = \dfrac{x}{1-|{x}|} $ How do I deal with the absolute value here? Thanks
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295 views

Calculating the trace of the product of two matrices

I have to calculated $\mbox{trace}(A^{-1}B)$ where $A$ is a symmetric positive definite matrix and $B$ is a symmetric matrix, very sparse with only two elements non zero. I want to find a way that I ...
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131 views

Inverse of identity plus scalar multiple of matrix

Given the matrix $M = ( I + \alpha D P )$, where $I$ is the nxn identity, $D$ is nxn symmetric and invertible, $P$ is nxn symmetric but not always invertible, and $\alpha$ is a scalar, is there a ...
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34 views

General formula for the inverse of the symmetric matrix $X$ defined as $a^{x_{ij}}$

Let $X$ be a $N\times N$ symmetric matrix with strictly positive entries $x_{ij}$. The inverse of $X$ is known. Let $0 < a < 1$ be a real number. Finally define $M$ as the matrix with entries ...
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52 views

Efficient diagonal update of matrix inverse

I am computing $(kI + A)^{-1}$ in an iterative algorithm where $k$ changes in each iteration. $I$ is an $n$-by-$n$ identity matrix, $A$ is an $n$-by-$n$ precomputed symmetric positive-definite matrix. ...
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29 views

Is inverse image of a point a set?

This is purely a matter of notation. Suppose $f(x) = x^2$. We say that $f(2) = 4$. We also say that $f(\{2\}) = \{4\}$. We can also say that $f^{-1}(0) = 0$ without much grief because there exists a ...
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What is the inverse of this function?

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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1answer
68 views

Inverse Laplace transform of $\large \frac{1}{s^2-As^{1.5}}$

Title says it all. How do I go about finding inverse Laplace transform of that expression? If it were complete exponents, I would have used partial fractions. But what to do with non integer ...
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1answer
34 views

Why is there a left inverse for an injective Function with the empty set as domain?

The fact that a function is injective is equivalent to the fact that there is a left inverse. Now consider $f:\mathbb{∅}\to \mathbb{A}$ where $\mathbb{A}$ is non-empty. Wouldn't the left inverse be ...
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39 views

Inverse laplace transform of complicated function

I have a function: $f(s)=\dfrac{(-HT/s)e^{-x*\sqrt{a/s}}}{\sqrt{a/s}+He^{-x*\sqrt{a/s}}}$ where s is frequency domain variable and H,T,a,x can be regarded as constants. How do I find inverse Laplace ...
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902 views

Inverse of function, containing a fraction

This is basic, I know, but I cannot seem to come up with the right answer. Find the inverse of the function: $$f(x)= \frac3{x+1}$$ My steps: 1. Convert f(x) to y $$y = \frac3{x+1}$$ Switch places ...
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44 views

How to solve the following equality

Is it possible to solve the following equation analytically for $\beta$: $$y'(A+\beta B)^{-1}y = \alpha,$$ where $A$ and $B$ are both positive-semidefinite and symmetric matrices (essentially, some ...
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174 views

Can someone please help with my inverse function and sets discrete math problem?

To save me some time writing everything out in latex, I'm adding a picture of the question and Ill try to explain what I understand for the problem. Just a heads up, I'm really not sure how to do this ...
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1answer
33 views

Exercise about linear operator

For $X$ Banach, I have to show that if $T\in\mathfrak{L}(X)$ and $||T||_{\mathfrak{L}(X)}<1$ then exists $(I-T)^{-1}$ and $$ (I-T)^{-1}=\sum_{n=0}^\infty T^n. $$ For the existence of $(I-T)^{-1}$ ...
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115 views

Is there a meaningful pseudo-inverse of a singular projection matrix?

Hello linear algebra experts. In my research I'd like to solve (or approximate) for B, in the form $ A = GBG $ where A and B are symmetric, square matrices and G is a symmetric, square, singular ...
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25 views

Chain rule with inverse function

In a proof, my professor shows: $ s = g^{-1}(u) $ $ ds = \frac{dg^{-1}(u)}{du} du $ , by the chain rule If I were to apply the chain rule to calculate ds, I would not get the du in the denominator. ...
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68 views

Find the inverse of a matrix in $GL(2\,,\, \Bbb Z_{11})$.

What are the necessary steps and reasoning for calculating the following matrix in GL(2,$\Bbb Z_{11}$): $M = \begin{pmatrix} 2&6 \\3&5 \end{pmatrix}$. I found the answer to be ...
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83 views

Let $f: A\rightarrow B$ and $g: B\rightarrow C$ be invertible maps, show that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

I am working on the following problem for my abstract algebra class, and I wanted to get some feed back to see if I am on the right track. Here is what I have so far. Let $f: A\rightarrow B$ and $g: ...
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18 views

Finding Inverse of Function With Two Instances of X

I need to find $f^{-1}(2)$ where $f(x) = 2 + x^2 + tan(πx/2)$ I know can substitute $f(x)$ with $y$ and swap $x$ and $y$: $$x = 2 + y^2 + tan(πy/2)$$ But I'm having trouble eliminating the tangent: ...
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98 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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51 views

is (I+P) invertible when row sum in P = 0

I have a $n$x$n$ matrix P where the sum of each row = 0 (the individual entries are real but can be negative). Clearly P is not invertible. Can we show that I+P is invertible? thanks
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29 views

Finding the integral of an inverse cosine function?

I've just been having trouble with this question: "Differentiate $xcos^{-1}x$ and hence find the integral of $cos^{-1}x$. Hint: Try using the substitution $u=1-x^2$." Finding the derivative wasn't ...
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35 views

Inverse of the sum of two orthogonal projections

I am trying to find out, if there is a formula for finding the inverse of the sum of two orthogonal projections. So basically my questions is: If $\left[\mathbf{A},\mathbf{B}\right]$ is full rank, ...
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73 views

Suppose R is an integral domain. Prove that $(a)=(b)$ if and only if $b = ua$ where $u$ is in $R^\times$

I am lost on this one. I'm still new to ring theory, as we're only a couple weeks into the course, but it's already well over my head. I know that $R$ is an integral domain, so the additive and ...
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47 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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44 views

Expressing an inverse trig function?

I just need a little help with this question: "Express cos$y$ in terms of cos $y/2$ and hence show that tan$^{-1} sqrt[(1-x)/(1+x)] = 1/2$ cos$^{-1}x$, for $0<x<1$." I can do the first part, ...
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21 views

Derivation of inner variations

In Giaquinta's and Hildebrandt's 1996, "Calculus of Variations 1", pages 147-148, they develop the definition of inner variations. They first fix $\lambda\in ...
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126 views

inverse of quadratic matrix form

I have an expression of the form: $ACA′$ where C is an invertible, symmetric and positive definite matrix. I'm trying to figure out if the expression above is invertible (or what additional ...
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1answer
209 views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
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74 views

Is there a name for an algebraic structure like this?

I'm self studying abstract algebra. I see that in rings there's no requirement for a multiplicative inverse. Is there something similar except with no requirement for an additive inverse. For ...
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61 views

Uniform continuity of inverse in only one variable

Let $f:[0,1]\times[0,1]\to \mathbb{R}$ be a (uniformly) continuous functions. Denote the image of $f$ by $D_f:=\{(x,y): x\in[0,1] , 0\leq y \leq f(x,1)\}$ $f$ is such that the section $f_x$, i.e. the ...
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82 views

Increase the diagonal entries of a positive definite matrix

Assume that we have a positive definite matrix $C$, and a positive definite diagonal matrix $\Lambda$. Are all the diagonal entries of $(C + \Lambda)^{-1}$ smaller than those of $C^{-1}$? In other ...