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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Primes and Inverses of an integer

I have the following question which I do not understand. Here it is: Consider the primes $5$, $7$ and $11$ as n. For each integer from $1$ through $n - 1$, calculate its inverse. I do not ...
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247 views

Proof of Vandermonde Matrix Inverse Formula

I'm working through Exercise 40 from section 1.2.3 of Knuth's The Art of Computer Programming volume 1, but am finding myself unable to produce a rigorous proof, and the one here is suspect and not ...
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1answer
42 views

If A = BC and B is invertible, then how does reducing “B to I” also reduce “A to C”?

If $A = B*C$, where $B$ is an inverse, use row-ops to reduces "$B$ to $I$" also shows that it will reduce "$A$ .. $C$". Big-Hint: Represent the row operations by a sequence of elementary matrices.
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30 views

Inverse trig functions [duplicate]

What would be the inverse function of $f(x) =x \cot \frac{\pi }{x}$ on the interval $ x\geq 3$? I don't know what to do about that pesky x in the front of the problem. Otherwise, the problem would be ...
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257 views

Show that A is invertible and that it is Lower Triangular.

Does anybody have a solution to the given word problem below? Let A be a lower triangular n x n matrix with nonzero entries on the diagonal. Show that A is invertible and and that A-inverse is lower ...
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115 views

Inverse of the Modified Bessel function

Is there any chance of having a formula or approximation to inverse the Modified Bessel function of the first kind? I mean to solve $I_M(x)$ with respect to x: $I^{-1}_M(x)$? Thanks in advance
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43 views

How to calculate the inverse of a known optical distortion function?

Assume I have the following lens distortion function: $$ x' = x(1 + k_1r^2 + k_2r^4) \\ y' = x(1 + k_1r^2 + k_2r^4) $$ where $r^2=x^2 + y^2$. Given the coefficients $k_1$ and $k_2$ I would need to ...
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157 views

convexity of inverse function

I have a question on the reverse of a convex function. Let $f(x)$ be a convex function. Is the reverse function, say $g(x)=f(x)^{-1}$, is necessarily a concave function ? Considering that such ...
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1answer
38 views

Factoring a matrix out of linear matrix equation

I'm having a bit of trouble following a solution in a textbook, one step in particular. I have the equation $(Z + tV)^{-1}$ where $Z$, $V$ are matrices and $t$ is a scalar. $Z$ is positive definite, ...
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2answers
57 views

Inverse of $r sin(\omega t) + v t$?

I am wondering if there is an inverse for this function, $x(t)=r sin(\omega t) + v t$. The inverse function theorem suggests that an inverse for this function does exist, although it may have to be ...
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58 views

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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1answer
90 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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1answer
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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1answer
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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1answer
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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1answer
34 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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56 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
73 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
178 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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196 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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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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98 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
116 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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99 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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34 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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1answer
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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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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135 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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1answer
55 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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2answers
75 views

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
69 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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1k 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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2answers
179 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
35 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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121 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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26 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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4answers
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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1answer
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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4answers
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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53 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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1answer
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 ...