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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2
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3answers
130 views

Find a constant so matrix is invertible

I am doing some exercises from my Linear Algebra textbook and i have come across an exercise which I don't quite understand. Every exercise is graded with numbers from [1] to [5]. [1] is meant to be ...
0
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0answers
26 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz

I have asked this question on mathoverflow also. (my question, I wasn't sure if its ok ask at another similar forum, on stack exchange, but I hope it would reach more people). It is well known how to ...
0
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1answer
15 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz

It is well known how to solve a Toeplitz system Ax = b, of a matrix A, n x n elements, ...
1
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0answers
28 views

Is it okay to perform the same row operation twice on opposite rows?

I am trying to find the inverse of the following matrix: 1 2 3 2 1 4 1 0 2 I draw the identity matrix next to it and start performing row operations. ...
1
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1answer
25 views

Inverse Permutations from $S_7$

Would someone mind giving an explanation of how to find the inverse permutation of: $(1 2 3 5 7)^{-1}$ in $S_7$? I am not quite understanding how to do this.
2
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1answer
67 views

Finding a binary operation on $\{1, \dots, n\}$ so that each $k$ has exactly $k - 1$ left inverses

What is an example of a binary operation on the set $\{1, \dots, n\}$ so that each element $k \in \{1, \dots, n\}$ has respectively $k-1$ left inverses? I have been trying various combinations ...
-1
votes
1answer
30 views

Inverse of rational function [on hold]

I need help with this question: Determine whether the given function is one-to-one, and if so, find the inverse: $$ f(x) = 5x + \frac{2}{x} $$ Wolfram says the answer is $\frac{1}{10}\left(x ...
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4answers
47 views

If $A$ is a $3\times3$ Matrics Then $\left |(2A)^{-1} \right |=?$ [closed]

If $A$ is a $3\times3$ matrics.And $\left | A \right | = -7$.Then what's the value of $\left |(2A)^{-1} \right |$ Please help to do this math easily.I tried a lot but still no idea come into my ...
1
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3answers
51 views

Slopes of inverse functions

I have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ...
2
votes
2answers
26 views

Choosing the right sign for inverse functions?

If I have to find an inverse function and through the algebra I get a $\pm$ sign how do I know which one to choose from if its in a given interval? For example a question asks: The function ...
0
votes
1answer
24 views

Is there an explicit formula for $\left(xx^T\right)^{-1}$ with $x\in\mathbb{R}^n\setminus\left\{0\right\}$?

Let $x\in\mathbb{R}^n\setminus\left\{0\right\}$. Obviously, $$A:=xx^T$$ is symmetric and positive definite. Hence, $A$ is invertible. Can we find an explicit formula for $A^{-1}$?
8
votes
4answers
130 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
1
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0answers
32 views

Inverse Laplace Transform using Hetnarski's Algorithm

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
0
votes
1answer
35 views

Inverting the differential operator $D^2-3D+2$ [closed]

I am trying to calculate $$(D^2-3D+2)^{-1}(xe^{3x})$$ that is, find a function $f$ such that $(D^2-3D+2)(f)=xe^{3x}$ where $D=\frac{d}{dx}$. Using inverse operator, I am getting an incorrect answer. ...
3
votes
1answer
50 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 $\dfrac{\sin x}{x}$?
0
votes
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
vote
3answers
34 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?
0
votes
4answers
60 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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votes
1answer
37 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.
3
votes
0answers
58 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 ...
0
votes
0answers
9 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} ...
1
vote
1answer
85 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 : ...
2
votes
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
votes
0answers
75 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
47 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 ...
1
vote
1answer
79 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
72 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
votes
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?
1
vote
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 ...
1
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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
29 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?
0
votes
0answers
32 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 ...
0
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, ...
0
votes
2answers
35 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 $f$?
1
vote
2answers
64 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 ...
0
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 $$\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)$$ ...
0
votes
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 ...
0
votes
1answer
26 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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votes
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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vote
0answers
15 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 ...
0
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
61 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
vote
1answer
46 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 $I = x^5 +x^3 + x = 3$ ?
2
votes
0answers
43 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\\ ...
0
votes
0answers
13 views

Differentiating integral by substituting inverse function

I have the following cost function that I wish to minimize with respect to $\alpha$: ...