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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29 views

Representation of the inverse of an variance-covariance matrix $\hat{\Sigma}^{-1}$

Given $T$ observed vectors $x_i\in\mathbb{R}^N, i\in\{1,\ldots,T\}$. Define $\hat{\Sigma}$ as the corresponding empirical covariance-matrix of the Observations $X=\left(\begin{array}{c} x_1' \\ ...
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1answer
40 views

Computing the inverse of linear transformations using matrices

For each of the following linear transformations T, determine whether T is invertible, and compute T-1 if it exists. (a) T: P2(R) $\to$ P2(R) defined by T(f(x)) = f ''(x) + 2 f '(x) - f(x). My ...
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1answer
14 views

Calculating the inverse of a continuous map for a certain interval in order to calculate the Perron-Frobenius operator.

Suppose we are observing chaotic continuos maps, the Perron-Frobenius operator $P$ satisfies: $P\phi_{n}(t) = \frac{d}{dt} \int_{f^{-1}([a,t])} \phi(x)dx$ I don't understand how for the shift map, ...
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1answer
29 views

What is the adjoint of an inverse matrix? [duplicate]

What is the adjoint of an inverse matrix? Is $(T^{-1})^{*} = (T^{*})^{-1}$?
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1answer
32 views

Permutations, compositions and associativity properties

Let n be a postive integer, and let σ : {1, . . . , n} → {1, . . . , n} be a one-to-one and onto map. Then σ is called a permutation on n elements. The set of all permutations on n elements is denoted ...
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4answers
55 views

Let $T$ be a linear transformation on a vector space $V$ ($\textrm{dim}\ V = n$). If $\textrm{rank}\ (T^2) = n$, is $T$ invertible?

For a linear transformation $T$ on a finite dimensional vector space $V$ ($\textrm{dim}\ V = n$). If $\textrm{rank}\ (T^2) = n$, is $T$ invertible? Also, is it guaranteed to have an eigenvalue?
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1answer
34 views

Series expansions of inverse polynomials

Suppose one is given a strictly monotonous polynomial, $$f(x) = \sum_{n=0}^N a_n x^n$$ So that for a given $y$ there exists a single real $x=f^{-1}(y)$. It would be nice* to be able to calculate the ...
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1answer
57 views

Inverse of $f(x) = 3x + \cos(x)$

Was hoping someone could help me find the inverse of $f(x) = 3x + \cos(x)$ The steps I took were: $y = 3x + \cos(x)$ $x = 3y + \cos(y)$ $x - 3y = \cos(y)$ $\arccos(x-3y) = y $ But I still have a ...
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1answer
24 views

Find $f^{-1}(g(x))$ if $f(x) = 2x + 1$ and $g(x) = x^{2}$

Question: Let $f$ and $g$ be defined as: $$f(x) = 2x + 1, ~~~~x \in \mathbb{R}$$ $$g(x) = x^{2}, ~~~~~~~~~~~~x \in \mathbb{R}$$ Find a) $~~f^{-1}(x)$ b) $~~f(g(x))$ c) $~~g(f(x))$ d) ...
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31 views

Why do the columns of the inverse of a matrix (defined as a linear operator) form an orthogonal basis in an inner product space?

Let V be a vector space over C and W be an inner product space over C with inner product <., .> and T:V --> W be a linear transformation. Find an orthogonal basis for V = R^3 with the inner product ...
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2answers
37 views

How to show $\text{rref }[\left.A\right|AB]=[\left.I_n\right|B]$?

For invertible $A^{n\times n}, B^{n\times n}$, how do I show that $\text{rref }[\left.A\right|AB]=[\left.I_n\right|B]?$ Tentatively: $\text{rref ...
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2answers
30 views

$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\frac{d}{dy}[\Phi(\Phi^{-1}(y))]}$?

If $\Phi(y)$ is a monotonic decreasing function is true that $$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\Phi'(\Phi^{-1}(y))}$$ If so, how? It works for $y = \Phi(x) = e^{-x}, \quad \Phi^{-1}(y) = ...
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1answer
25 views

how to do inverse laplace of $(s^2+1)/s^4$?

how to do the inverse laplace of $(s^2+1)/s^4$? the answer is $(t^3/6)+t$ but I do not know how to derive it.
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3answers
59 views

Find the inverse of $f(x,y) = (x+3y,3x+y)$

Given the function $f : \mathbb{R}^2 \to \mathbb{R}^2$ as $f(x,y) = (x+3y,3x+y)$. Find $f^{-1}$ .( Assume $f$ is a bijection) I know how to find $f^{-1} (x) = (3x+2)$ or anything with one ...
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1answer
99 views

Inverse of $f(x) = a \left(1 + \frac{c}{(1+x^b)^{-\frac{1}{b}} - c}\right) \cdot (1+x^{-b})^\frac{1}{b}$?

How can one find the inverse of $$ f(x) = \mathrm{a} \left(1 + \frac{\mathrm{c}}{(1+x^\mathrm{b})^{-\frac{1}{\mathrm{b}}} - \mathrm{c}}\right) \cdot (1+x^{-\mathrm{b}})^\frac{1}{\mathrm{b}} $$ with ...
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1answer
51 views

Inverse function table

I am required to create a table of values (like the one above) for h-1(x). Because x is ordered, i am just wondering, would the two tables would be identical? I just feel a little insulted that's ...
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19 views

inverse fourier transform of unit impulse function of omega

What is the inverse fourier transform of the unit impulse function of omega. Sorry I've not got the symbol in my phone. It Should looks like §(W).. Sorry for the special symbols.
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1answer
28 views

Inverse of elementary functions

which may be two right inverse of: 1) $h:\Re \rightarrow [0,\infty) $ defined by $h(x)=|x|$ 2) $k:\Re \rightarrow [1,\infty)$ defined by $k(x)= e^{x^2}$
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2answers
58 views

$\sqrt{4x -3}$ injective? Bijective? Inverse?

I'm shown part of the function $g(x) = \sqrt{4x-3}$. Is it injective? I said yes as per definition if $f(x) = f(y)$, then $x =y$. Is this right? Under what criteria is $g(x)$ bijective? For what ...
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6answers
51 views

Finding the inverse of a number under a certain modulus

How does one get the inverse of 7 modulo 11? I know the answer is supposed to be 8, but have no idea how to reach or calculate that figure. Likewise, I have the same problem finding the inverse of 3 ...
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1answer
28 views

Functions problem: surjectivity and direct and inverse image theory

I need some help with this problem, if sombody could give me any idea of how to solve it (not the solution itself, but it would be better) I will appreciate it: for a function $f: A → B$, prove $ ∀ Z ...
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3answers
31 views

Verify matrix identity

The question: Show that if $A$, $B$, and $A+B$ are invertible matrices with the same size, then: $$A(A^{-1}+B^{-1})B(A+B)^{-1} = I$$ I began by multiplying the first $A$: ...
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1answer
156 views

Analytic inverse of $f(z) \neq 0, f(0) = 0, f'(z) \neq 0 $ within minimum modulus on boundary.

Suppose $f(z)$ is analytic on closed disk of radius $r$ and $f(0)=0$, $f'(z) \neq 0$. Show that $f$ has an analytic inverse on $\{|z| \leq m\}$ where $m$ is the minimum of $|f(z)|$ on $\{|z| = r\}$. ...
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1answer
33 views

Inverse Laplace Transform with time delay and extra factor

I am attempting to solve a PDE $$y_{tt} = y_{xx}, -\infty < x < 0,\ t > 0$$ with boundary conditions $$ y_x(0,t) = k(t),\ y(x,t) \rightarrow 0\ \mbox{as}\ x \rightarrow -\infty,\ y(x,0) = 0,\ ...
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0answers
15 views

Closed form expression for the S-transform of a random variable?

I'm trying to compute the S-transform as described in this review article on random matrix theory (section 2.2.6). They define it as $\Sigma_X(\gamma) = -\frac{\gamma+1}{\gamma}\eta_X^{-1}(1+\gamma)$ ...
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2answers
40 views

Computing $\arctan x$ in terms of a certain collection of other functions

I know that $$\tan(x) = \frac{\sin(x)}{\cos(x)}.$$ Does this relationship hold in the inverse in any form? For example: atan(x) = asin(x) / acos(x), or atan(x) = acos(x) / asin(x), or atan(x) = ...
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3answers
52 views

Inverse function of $x^x$

How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried ...
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1answer
26 views

Showing topological properties of a function

Let $f$ be a function from a set $X$ into a set $Y$. prove: i) the function $f$ has an inverse if and only if $f$ is bijective ii) let $g_1$ and $g_2$ be functions from Y into X. If $g_1$ and $g_2$ ...
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Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$?

Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$ In other words, To show $f$ and $g$ are inverse, is it enough to show $(f\text{ o }g)(x)=x$? I have never witnessed a case in which the ...
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0answers
18 views

inverse of semi infinite toeplitz matrix

I have a semi infinite toeplitz matrix of the form $ A=\left(\begin{array}{ccccc} A_{0} & A_{1} & 0 & 0 & \cdots\\ A_{-1} & A_{0} & A_{1} & 0 & \cdots\\ 0 & A_{-1} ...
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0answers
23 views

“Fix” an Inaccurate Matrix Inverse

I have $A$, a poorly-conditioned (but not singular) matrix. The matrix $B\approx A^{-1}$ (currently computed using double-precision Gaussian elimination), is rather further from the true $A^{-1}$ ...
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30 views

Show if $(a,p)=1$ there is a unique inverse of $a$ modulo $p$

In a proof of Wilson's theorem, I read this identity and just wondered how to prove it: When $1\leq a\leq p-1$, we have $(a,p)=1$, so there exists a unique $\overline{a}$ with $a\overline{a}\equiv ...
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3answers
97 views

When does a function have an inverse?

I have been told that a function has an inverse if it is one-to-one or injective, but how can we rigorously prove this? I have been struggling to find a proof for days.
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1answer
78 views

Inverse $z$ transform - contour integration

Here is my task: Find inverse $z$ transform of $$X(z)=\frac{1}{2-3z}$$if $$|z|>\frac{2}{3}$$ using definition formula. I found that $$x(n)=\dfrac{1}{3}\left (\dfrac{2}{3}\right ...
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1answer
40 views

Solve the triangle using the law of Cosine

$a = 20$, $c= 20$, $B = 30^\circ$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$ Using the law of cosine: $b^2 = a^2 + c^2 - 2ac\cos(B)$ \begin{align*} b^2 & = 20^2 + 20^2 -2(20)(20)\frac{\sqrt{3}}{2}\\ ...
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0answers
20 views

Deriving the inverse of RGB to HSV transformation

Today we had the conversion from RGB to HSV coordinates and vice versa in a multimedia systems exam. And I was puzzled between the conversion. RGB and HSV are of course the color spaces. The ...
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2answers
34 views

If I have two functions defined from $\mathbb{R^3} \to \mathbb{R}$, can they be inverses?

I am taking an economics class and I am not getting some straight answers about the conditions under which I can say that a map from $$\Bbb{R}^3 \rightarrow \Bbb{R}$$ can be reduced to a map from ...
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2answers
38 views

Find an example of continuous but not increasing function whose inverse function doesn't satisfy the Inverse Function Theorem

I have to find an example of a function $f:[a,b]→R$ which is continuous, but not strictly increasing, such that no inverse function $f^{−1}$ satisfy the property of the Inverse Function Theorem.
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1answer
26 views

Finding possible inverses of a modulo function

I know how to find $one$ inverse via the euclidean algorithm, but I can't figure out how to find more of them. For example: Find an inverse $x$, of $57$ $modulo$ $100$ Or an $x$ such that $57x ≡ 1$ ...
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2answers
33 views

Differentiating inverse hyperbolic function

I am trying to differentiate $\tanh^{−1}\left(x/(1 + x^2)\right)$, but am finding it difficult understanding what to do. I think you have to place the differential of the angle of the hyperbolic ...
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1answer
48 views

Different Inverse Approach

As it is known, we use inverse (Gauss Elm, Jordan...) or pseudo-inverse methods (LU, SVD, Chol, QR...) to solve linear equation namely $ A*x=b$ when $A$ is $[m,n]$ and $b$ is $[1,n]$ matrix. These all ...
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3answers
48 views

Functions - Inverses of graphs.

The question reads: sketch the graph of y=-3-x along with its inverse. From calculating the equation of the inverse graph, I come to y=-3-x, using the swap method. I then tried to plot both graphs ...
2
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1answer
39 views

Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
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1answer
14 views

for $1 \geq x \geq 0: {2x^2\over{(2+x)}} \leq y \Rightarrow x \leq \left(\frac{3}{2} y \right)^{1/2}$

So what I did is prove that $f(x) := {2x^2\over{2+x}}$ is increasing and then invert $f$ on $[0,\infty]$ this yields $(f\restriction_{[0,\infty[})^{-1}(y) = \frac{1}{4}(y+\sqrt{y}\sqrt{y+16})$ and ...
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0answers
40 views

Difficult examples of invertible, differentiable functions

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
2
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0answers
49 views

Inversion of a pairing function

I was reading this question on this site and I saw that the following pairing function was mentioned (a modified version of Cantor function): $$\langle x, y\rangle = x * y + ...
5
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1answer
143 views

Find Functions That Can Be Inverted from Their Sums

I have the following situation:$$ f_1(x_1) + f_1(x_2) + f_1(x_3) + \cdots + f_1(x_n) = c_1\\ f_2(x_1) + f_2(x_2) + f_2(x_3) + \cdots + f_2(x_n) = c_2\\ \vdots\\ f_n(x_1) + f_n(x_2) + f_n(x_3) + ...
4
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3answers
57 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 ...
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1answer
37 views

Exponential Function Equation and inverse Pre-Cal

I am not completely sure if I wrote the equation correctly. For A I wrote: $m(t)=100(b^x)$ Not sure it is correct...but how do I find the inverse? That doesn't make sense to me. Do I use log?
5
votes
1answer
124 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...