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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53 views

Inverse Laplace transform of $\operatorname{arccot}(s)$, $\arctan(s)$

How would one find inverse Laplace transforms of $\operatorname{arccot}(s)$ or of $\arctan(s)$ without knowing in advance that this is related to $\dfrac{\sin x}{x}$?
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1answer
33 views

Find inverse $f^{-1}$ of a function $f(x,y)=(x-y,x-10y)$ [duplicate]

I know how to find inverse function if the given function is in the explicit form. Could someone show on this example how to find $f^{-1}$? Thanks for replies.
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2answers
36 views

For given $t$ and $x$ and $y$, is there at least one $f$ such that $\cos ft = x, \sin ft =y$?

Suppose that $t$, $x$ and $y$ are given and are all in $\mathbb{R}$. Is there always at least one $f$ such that $\cos ft = x, \sin ft =y$? Edit: OK I forgot to add that given $x$ and $y$ are such ...
1
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3answers
36 views

Solve for $x$ from an equation containing inverse trigonometric functions

How to solve the following for $x$? $$ \sin^{-1}\left(\frac{2a}{1+a^{2}}\right)+ \sin^{-1}\left(\frac{2b}{1+b^{2}}\right)= 2 \tan^{-1}(x ) $$ What conditions apply?
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4answers
62 views

If $\arctan(x)+\arctan(y)+\arctan(z)=\pi/2$ how to show that $xy+yz+zx=1$? [closed]

If $\arctan(x)+\arctan(y)+\arctan(z)=\pi/2$ how to show that $xy+yz+zx=1$ ?
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1answer
38 views

Invertibility Proof for matrix

Suppose that A is a square matrix that satisfies $A^n=0$ for some positive integer n. Show that $I-A$ is invertible and $(I-A)^{-1}=I+A+A^2+...+A^{n-1}$. Not sure how to start the problem.
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61 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
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0answers
12 views

Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation

Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of: $$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} ...
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1answer
88 views

What does $\; \lim\limits_{x\to\infty} \arccos x =i\infty \;$ mean?

Is there somone who can show me what $\; \lim_{x\to\infty} (\arccos x) =i\infty \;$ means? Does it meant that limit does not exist? $\:$ If yes, how can one prove that limit does not exist? Note : ...
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2answers
20 views

Does every $\mod p$ have at least one element with a non-identical inverse?

Does every mod p have at least one element with a non-identical inverse? I very much suspect this is true, but how can I prove it? For example, in mod 5, some elements have inverses that are not ...
4
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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
votes
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
81 views

Why determinants can be used to find inverses of $2 \times 2$ matrices [closed]

In linear algebra, you can find the inverse of a square matrix of dimensions $2\times 2$ by multiplying all the elements of the matrix - where the matrix is altered to have elements $a_{12}, a_{21}$ ...
2
votes
5answers
76 views

Value of $x$ in $\sin^{-1}(x)+\sin^{-1}(1-x)=\cos^{-1}(x)$

How can we find the value of $x$ in $\sin^{-1}(x)+\sin^{-1}(1-x)=\cos^{-1}(x)$? Note that $\sin^{-1}$ is the inverse sine function. i'm asking for the solution x for this equation Pls workout the ...
0
votes
1answer
19 views

How to compute the eigenvalues?

Suppose $W=(X'X + kI)^{-1}$ and $Z=(I + k(X'X)^{-1})^{-1}$, $k>0$, and suppose also that $\lambda_i$ are eigenvalues of $X'X$. How to get the following conclusions about their eigenvalues. The ...
3
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3answers
37 views

Inverse function of $f(x,y,z) = (xy-z^2, x+z)$?

How do you determine the inverse function $f^{-1}: \mathbb{R}^2 \to \mathbb{R}^3$ of $f: \mathbb{R}^3 \to \mathbb{R}^2 , f(x,y,z) = (xy-z^2, x+z) $ ? Or to put it into a bigger context: ...
0
votes
1answer
25 views

Finding the Inverse of this function

Im trying to find the inverse of this function $$x \mapsto\frac {113^x - 1}{112}\def\comment#1{}\comment{(pow(113.0, x)-1.0)/112.0} $$ But it always turn up incorrect. Can someone point me in ...
1
vote
1answer
28 views

Finding the inverse of a function in two variables

I have a function $f$ on the integers in $[-180,180)\times [-90,90)$ defined by $$f(y,x) = y + 360 x$$ I would like to find the inverse function. How can I do this?
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2answers
49 views

Inverting an arbitrary integral

$$r(x) = \int_{x_\min}^x f(y)\, dy$$ I would like to obtain an inverse for this such that I have $x(r)$. Is this possible? I saw this post before, however my function has a $y$ involved which makes ...
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0answers
24 views

Ideas for expressing the inverse of matrix quadratic form $CAC^T$

I want to find an expression for the inverse of the matrix system $Z=CAC^T$, where $A \in \mathbb{C}^{n \times n}$ is block diagonal with dense blocks, and $C \in \{-1,0,1\}$ with dimension $m \times ...
3
votes
1answer
38 views

$ (x x^T)^{-1}$, efficient matrix inversion for matrix composed as product of a vector with itself?

Given a vector $x$, is there an efficient way of computing $(x x^T)^{-1}$? I mean without first computing the matrix $(x x^T)$ and then applying matrix inversion techniques to it?
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0answers
40 views

Inverse of a toeplitz matrix with fft based methods

I have a covariance matrix, Q and I need to find out Q^-1. Here, Q is a Toeplitz matrix. Now, I want to calculate the inverse of the matrix with fft based methods rather than the conventional ones ...
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votes
0answers
21 views

Finding the Inverse of Polynomial Equations (Approximatly)

Assume one is given a set of two equations of the form: $$x(u,v) = u + a_1 u^2 + b_1 u v + c_1 v^2$$ $$y(u,v) = v + a_2 u^2 + b_2 u v + c_2 v^2$$ And one would like to find the inverse functions, ...
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2answers
42 views

What is the domain of an inverse function?

If $f:X \to Y$ then if the inverse exists, is the domain the range of $f$ or the codomain of $f$?
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2answers
68 views

How to show that Id + skew matrix is invertible [duplicate]

How does one prove that the sum of the identity matrix and a given matrix $A$, when $A$ is an antisymmetric matrix, is invertible? I tried to show that the rows / cols are linearly independent, or ...
0
votes
1answer
25 views

Inverse of function containing modulation and flooring

I have a function $f: \mathbb{N} \rightarrow \mathbb{N}$ defined as: $$f(x) = ((x \bmod 9) + 1) \cdot 10^{\lfloor \frac{x}{9} \rfloor}$$ It seems to be injective, but I'm not sure about it being ...
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votes
1answer
19 views

Compute $(df)_a$ in chart $\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)$

Suppose that for a submanifold $H$ of $\mathbb{R}^3$ we have two charts $$\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)$$ ...
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1answer
15 views

Inverse of a linear transformation

What is the inverse of the following linear transformation? $T^{\theta}:R^2\rightarrow R^2$ a reflection in the line through the origin which forms an angle $\theta$ with the $x$-axis. I ...
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votes
1answer
30 views

Does “f : A → B” need to be one-to-one and onto so that if Y ⊆ B, then the inverse image of Y under f and the image of Y under f-1 are equal?

I was solving a problem in section 5.4 of "How to Prove it Right" by velleman. Below are the problem and my answer. According to my inspection, $f$ didn't need to be one-to-one and onto. Did I miss ...
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1answer
58 views

Inverse modulo without brute force [closed]

I have this piece of code and I want to know 'x' before the loop without brute force. Is there a way to do an inverse modulo or something? ...
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0answers
16 views

Inverse of a Bijective Bivariate quadratic function or polynomial

I am looking for some general way to invert a bijective quadratic polynomial of the form $$ f(x,y)=A_0x+A_1x^2+Axy+B_0y+B_1y^2+Byx $$ where the coefficients may or may not be in the same ring as the ...
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votes
0answers
13 views

Inverse of pairing function

I am looking for the inverse of the unordered pairing function: $$ \langle x,y\rangle = xy + \left\lfloor \frac{\big( |x-y|-1 \big)^2}{4} \right\rfloor $$ where $x$ and $y$ are positive integers. ...
2
votes
1answer
66 views

Find the inverse $f(x) = 2x^2-8x, x>2 $

$$ 2x^2-8x, x>2 $$ What is the best way to solve this problem. $$x = 2y^2-8y $$ $$x = y (2y-8) $$ do I divide both sides by $y$ so as to solve for $y$? Help
1
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1answer
47 views

Assume that f is a one to one function: If $f(x) = x^5 + x^3 +x$ , find $f^{-1}(3)$ and $f(f^{-1}(2))$

If $f(x) = x^5 + x^3 +x$ , find $f^{-1}(3)$ and $f(f^{-1}(2))$ How do I go about solving this? For example, since I am giving f inverse should $I = x^5 +x^3 + x = 3$ ?
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votes
0answers
45 views

Norm of the inverse of a tridiagonal

Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric) $$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ ...
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0answers
13 views

Differentiating integral by substituting inverse function

I have the following cost function that I wish to minimize with respect to $\alpha$: ...
0
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1answer
37 views

Formal name for the coordinate values of the pushforward of the inverse metric on an embedded manifold?

What is the formal name of the following object: \begin{align}\tag{4} \Delta^{\alpha \beta} = \dfrac{\partial y^\alpha}{\partial x^m} g^{mn} \dfrac{\partial y^\beta}{\partial x^n} \end{align} where ...
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votes
0answers
12 views

Inverse Variation Graph

How would I graph an inverse variation in which y varies inversely as x and y=2 when x=7? I know that I have to follow the xy formula. So far, I found xy=14, x(2)=14, x=7. How do I make a table and ...
2
votes
1answer
46 views

Inverting the Radial Distortion

Overview The problem is perhaps a very easy one for a trained mathematician. As I am not a mathematician, but instead a researcher in general problem solving, I am reaching out to those who know more ...
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votes
2answers
60 views

Inverse of the Toeplitz matrix

I am working on the inverse of the sum of an identity matrix and a Toepltz matrix, and trying to find the formula for the (1,1) element of the inverse. For example, Assume $c$ is a nonzero constant, ...
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votes
0answers
38 views

What is $dx/dF(x)$ where $F(.)$ is a continuous, increasing function.

I was wondering if it is possible to find $dx/dF(x)$, that is, the derivative of $x$ with respect to $F(x)$, which is an increasing, continuous function. Does it involve finding the derivative of the ...
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0answers
36 views

Inverse of $R^T D R$ where $R$: rectangular and $D$: diagonal

Is there any formula for the following triple product: $$(R^T D R)^{-1}$$ where $R$ is rectangular and $D$ is diagonal? The real situation is like this. I have the equation $$Ax = b$$ which is a ...
2
votes
2answers
40 views

How is $X(X^{\prime}X)^{-}X^{\prime}$ symmetric?

For a matrix $X$, a generalized inverse of $X$ is any matrix $Y$ such that $XYX = X$. We use $X^{-}$ to indicate a generalized inverse of $X$. Suppose $X$ is a matrix. $X^{\prime}$ denotes the ...
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votes
3answers
24 views

Finding the inverse of an exponential function

I have this function: $$F_X(x) = \frac{3e^{2x}}{4} + \frac{3e^{4x}}{8} - 0.1$$ of which I am trying to find the inverse function, as in $u = F_X^{-1}(x)$. I made it to this form: $$u= ...
3
votes
1answer
36 views

Invertible matrix over a ring and its eigenvalues

Eigenvalues and invertible matrices for fields and vector spaces: Let $K$ be a field (so $K^n$ is a $K$-vector space) and let $A \in K^{n\times n}$ be an $n\times n$-matrix. Then we have the following ...
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votes
5answers
51 views

Find the inverse function of $g(x)=(x-2)(x-4),\; x≥3$.

Find the inverse of the following function, stating its domain. $$ g(x) = (x-2)(x-4), \quad x≥3. $$ I try to find the inverse function, but I can't eliminate $x$ in my method.
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votes
1answer
18 views

Left and Right Inverses with semigroups

Having the semigroup $(F,\circ)$ where $F=\{f: f: \mathbb{N}\to \mathbb{N}, \mathrm{Dom}(f) = \mathbb{N}\}$. The identity $e∈F$ is the function $e(n) = n$, define the function $g(n) = m$ if ...
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votes
2answers
46 views

Inverse of a function $xe^x$

How should I proceed about finding the inverse of the function $xe^x$? I have been wondering about it for a long time and can't think of anything to do.
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vote
0answers
55 views

Is there a closed form expression for $(A^T\Sigma A)^{-1}$ when $A$ is not square?

I need to find the inverse of the matrix $A^T\Sigma A$. Matrix $A$ has dimensions $5\times 2$. Matrix $\Sigma$ has dimensions $5\times 5$, and it is symmetric and positive-definite. I need to ...
1
vote
1answer
61 views

$f '(x) = -f(x)$ and $f(1) = 1$, Solve for $f(2)$

I am honestly not even sure how to start this problem... My first thought was that $f(2) = 2$ ... But now I don't even know where to go from there.