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 ...

learn more… | top users | synonyms

0
votes
3answers
20 views

Why does $\frac{1}{6e^{2y}}=\frac{1}{2x-8}$ in this context?

This is the context: I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
1
vote
2answers
20 views

Asymptotes of $\arctan (2x)$

My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, ...
0
votes
1answer
9 views

Geometric progression with reverse order

I have the following problem: Find three positive numbers which have the sum of $70$ and create a Geometric progression ($q>0$, increasing). Their inverse sum equals to $4/70$. Thank you!
0
votes
2answers
48 views

Is a factorable polynomial invertible?

The reason there exists no quintic formula that finds the roots of a quintic polynomial is simply because some quintic polynomials are irreducible. But reducible quintic polynomials may be invertible ...
-4
votes
1answer
60 views

What is the inverse function of $y=x^2 + 3x +2$? [on hold]

What is the inverse function of $f(x)=x^2 + 3x +2$? Please show your solution method and demonstrate that $f(f^{-1}(x))=x$
2
votes
3answers
49 views

Can you inverse a funcion by rotating it?

In school i sometimes run on some excercises where you need to calculate something that has an inverse function in it but you cannot find the inverse and you need to work your way around it. I know ...
1
vote
1answer
22 views

Inverse of the composition of two functions

If I have a composition of two functions: $$y = f(g(x),h(x))$$ where both $g(x)$ and $h(x)$ are readily invertible, can I find the inverse of the composition? i.e.: Can I find $x = f^{-1}(y)$? I ...
1
vote
1answer
15 views

Invariant under $x \rightarrow 1/x$?

I started thinking on the following problem. I am interested in finding complex functions of a complex variable such that $\phi(z)=\phi(z^{-1})$ So far, all I could come up with was a family of ...
4
votes
1answer
28 views

Does $\sin^{-1}x$ has a vertical tangent

I read that the function $f(x)$ has a vertical tangent at $x=a$ in the domain of $f$ if $$f'(a^-) \to +\infty$$ and $$f'(a^+) \to +\infty$$ Or both approach to $-\infty$. But for $f(x)=\sin^{-1}x$ ...
3
votes
2answers
38 views

For which values of $a,b$ is the matrix invertible?

I am trying to figure out the below question: 15. For which values of the constants $a$ and $b$ is the matrix $$A = \left[\begin{array}{cc} a & -b \\ b & a \end{array}\right]$$ ...
1
vote
0answers
15 views

Integral inversion

Say I know this function $$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$$ where $m(x)$ is a Fourier transform of an infinitely differentiable real function, whose maximal ...
0
votes
2answers
31 views

Can the cross product of two non-invertible matrices be invertible?

To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible? Just used to help understand a Linear Transformation assignment question, don't ...
1
vote
2answers
45 views

condition number of matrix plus constant times identity

I saw this post on the eigenvalues of a matrix plus a constant times the identity matrix. Say $A$ is an $n\times n$ matrix (real and non-singular) with eigenvalues $\lambda_1,\ldots,\lambda_n$, then ...
0
votes
2answers
25 views

Inverse image of the set $[−1, 4)$ under $f : x \mapsto -x^2$

I have an issue with determining the inverse image of a set. I cannot understand anything about it. I've got a simple exercise here, could someone here show me how the inverse image works and more ...
1
vote
1answer
36 views

What is $\ln(e^x -4) $, solving for the inverse?

What is $\ln(e^x -4) $, solving for the inverse? I know $\ln(e^x)$ is just $x$, but I don't know what to do with the 4.
0
votes
0answers
18 views

Finding Inverse of a matrix using elementary transformations

So I have to find the Inverse of A. $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 3 & 4 & 3 \\ \end{bmatrix} $$ By using elementary row or column transformations.. The ...
1
vote
0answers
35 views

Numerical Algorithm for $n \times n$ Matrix Inverse

I have to write a C program in which I have to compute the matrix inverse of a $n \times n$ matrix. Is there a convenient iterative process that I can use to do that? All I see is the co factor method ...
-2
votes
1answer
29 views

Finding the inverse of a function $f(x_1,x_2) = x_1/x_2$? [closed]

I'm trying to understand the inverse of the function $$y_1 = \frac{x_1}{x_2} \to x_1 = \sqrt{y_1y_2}$$ and $$y_2 = x_1 x_2 \to x_2 = \sqrt{\frac{y_2}{y_1}}$$
1
vote
1answer
16 views

Special Case Linear Solvers

I, and friends of mine, are interested in matrices which can be inverted / solved easily (i.e. in less than O(n^3)). I started to put together a github page dedicated to it and so far have identified: ...
-1
votes
0answers
26 views

Do $p=2617$ and $q=3571$ have modular multiplicative inverse with $e = 17$?

I need multiplicative inverse of $17 \mod \left(\phi(p) \cdot \phi(q)\right)$. They are both prime, the totient of the product is $2616 \cdot 3570 = 9339120$. But $17$ is a factor of $9339120$, ...
1
vote
0answers
38 views

Any suggestions on how to find the inverse of this function?

What is the inverse of $$ d (X) = \frac {C \left( 1 - \frac b C \right)^2} {2 \left( 1-X \frac C b \right)} + \frac {X^2} {2 v (1-X)} - 0.65 \sqrt[3]{ \frac c {v^2}} X^{2+ \frac b C}$$ where $b, C, ...
-3
votes
1answer
67 views

Find 2X2 matrices A and B that are not invertible but A+B is invertible. [closed]

How would I find the solution to this problem?
0
votes
0answers
10 views

Invariance of Frobenious norm under transformation.

Can we say for every invertible square matrix $\mathbf{P}$, $\Vert\mathbf{X-B}\Vert_F^2=\Vert\mathbf{P^{-1}(X-B)}\Vert_F^2$. And does this hold true for non-square matrix $\mathbf{P}$ under some ...
0
votes
4answers
72 views

How does $1 + \tan^2x = 1/\cos^2x$?

I am unable to see why $$1 + \tan^2 x= 1/\cos^2x$$ I have looked into the topic anad I am familiar with the reciprocal ratios of cosec, sec, and cot. but cannot derive how this statement makes sense. ...
0
votes
1answer
16 views

Having a holomorphic function $h$ that's the inverse of a function $f$, it's also the inverse for a continuation of $f$

Let $\gamma: [0, 1] \to \mathbb{C}$ be a (continuous) path, $\gamma(0) \in D$, $(f, D)$ a tuple of a holomorphic function $f: D \to \mathbb{C}, D \subseteq \mathbb{C}$ a simply connected open set. Let ...
0
votes
2answers
40 views

Proving matrix properties: [closed]

Prove: (i) $A(I+BA)^{-1}=(I+AB)^{-1}A$ (ii) $(I+AB)^{-1}=I-A(I+BA)^{-1}B$ (i) Consider $A(I+BA)=(A+ABA)=(I+AB)A$ Taking inverse on both sides (invert) ...
0
votes
0answers
37 views

Using Chinese Remainder Theorem to find an integer $x$ for which $ x\equiv 3\pmod 4 x\equiv 5\pmod 9 x\equiv 10\pmod {35} $

Hello I have got problems with understanding the reduction method in CRT. We have got system like this $$x\equiv 3\pmod 4$$ $$x\equiv 5\pmod 9$$ $$x\equiv 10\pmod {35}$$ There is a way to do this ...
2
votes
1answer
47 views

Inverse of matrix with particular structure

I have a square invertible matrix $A=[c, c^2, c^3 \dots c^n]$ where $c \in \Bbb R^n$. Are there any known fast tricks for inverting it? Edit: $c$ is a column vector and raising it to a power is to ...
0
votes
0answers
30 views

What approximations for the Gamma function's inverse appear to work 'best'?

So I was wondering how we approximate the inverse of the Gamma function, where I tried a few methods: Lagrange inversion theorem: $$\Gamma^{-1}(z)=a+\sum_{n=1}^{\infty}\lim_{w\to ...
1
vote
1answer
27 views

Inverse trigonometric expansion related question

I know expansions for $\sin^{-1}(x)+\sin^{-1}(y)$, but does there exists any expansion for $\sin^{-1}(x \pm y)$ if not then what is the reason?
0
votes
2answers
61 views

How do I prove that $(ABC)^{-1} = C^{-1} B^{-1} A^{-1}$ [closed]

Please help me answering this problem! thank you :) Prove that for any nonsingular matrices $A$, $B$, and $C$, the equation $$(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$$ holds. (Hint: Assume $D$ is the ...
0
votes
2answers
27 views

what condition of A makes transpose(A)*A nonsingular?

What contidion of A makes $$A^TA$$ nonsingular? If so, that is $$A^TA$$ is non-singular than a unique solution exists.
0
votes
0answers
23 views

partially ordered group, does x=-x imply x=0?

I have just a simple question: Let (G,+) be a partially ordered Abelian group. Does x = -x imply x = 0 ? If the answer is yes, then how could i prove it? If the answer is no, then a ...
0
votes
1answer
28 views

Find $\sinh^{-1}x$

The hyperbolic sine function, $\sinh(x)$ , is defined by the equation: $$ \sinh(x) = \frac {e^x-e^{-x}} {2}$$ Find a formula for its inverse, $$ \sinh^{-1}(x) $$
1
vote
1answer
21 views

Why arsin function has range $[-\pi/2,\pi/2]$ [duplicate]

While studying in P.75 of inverse trigonometric functions it tells we have to restrict our domain before finding the inverse.But I can't get why we choose $[-\pi/2,\pi/2]$?Why can't we choose ...
1
vote
1answer
12 views

Ensure that for each number in specific space there is inverse

Let say I want to find the Inverse number of some serial number. ( 9 digits number .. its can be an ID). And let say we want to find the inverse in $\mathbb Z_{1000000123}$ ( for example ) How I can ...
0
votes
1answer
83 views

How to convert $(A+\lambda E)^{-1}$?

Here is one of the most famous equation called Sherman–Morrison formula (1951) when we want to get an inverse matrix. ...
0
votes
1answer
65 views

Getting $B$ from $A = M^t B M$ without inverting $M$

I have got three matrices: $A$ (dimension $n \times n$), $B$ (dimension $m \times m$) and $M$ (dimension $m \times n$). We have $m > n$. This is the relation between these three matrices: $A = M^t ...
6
votes
0answers
148 views

How to invert this expression involving $\tanh^{-1}$?

I've got the expression: $ x = \tanh^{-1}(p) - \sqrt{\frac{2}{3}} \tanh^{-1}\left( \sqrt{\frac{2}{3}} p\right) $ How can I invert this function so I have a function $p(x)$? I thought about using ...
0
votes
2answers
87 views

How to find inverse of below function $y=2^x+3^x ,\ y^{-1}=?$ [closed]

I need to find inverse of below function: $$y=2^x+3^x ?$$
1
vote
0answers
20 views

Why is the range of inverse cotangent only positive? [duplicate]

Is it because of conventions or some practical reason? Some websites say it is conventions while other websites say it is because of making it a function. Thanks!
2
votes
0answers
17 views

Computing one-sided inverse of a matrix over some finite field

Let $M$ be a $k\times n$ matrix with $k < n$, and assume that $\text{rank}(M)=k$. Over $\mathbb{R}$, one can compute a right inverse of $M$ as follows: $$M_\text{right}^{-1} = M^T(MM^T)^{-1}$$ ...
2
votes
2answers
31 views

Determinant of complex matrix with almost constant lines

Let $0\neq c\in\mathbb{C}$. Take the matrix $$A_C=\begin{pmatrix} n&c&\dots&c&c \\ c&n&c &\dots & c\\ c &c & n &c &\dots\\ \vdots ...
1
vote
2answers
41 views

Eigen values of inverse matrix [closed]

Given a matrix B how to find the eigenvalues of its inverse?
3
votes
4answers
101 views

explanation of $ \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} $?

I'm studying about derivative of inverse function. The teacher in the video (https://www.youtube.com/watch?v=3ReOtNCYuBw) (at 9:00 minute) said this if a differentiable function, f has an inverse, ...
2
votes
3answers
40 views

Bezout's Identity proof and the Extended Euclidean Algorithm

I am trying to learn the logic behind the Extended Euclidean Algorithm and I am having a really difficult time understanding all the online tutorials and videos out there. To make it clear, though, I ...
1
vote
0answers
30 views

Invertibility, inverse, and line weight of big circulant matrices

I am generating a random square sparse binary circulant matrix, defined by its first row. The length of the matrix is 9857 bits, and each line contains 71 ones, the rest are zeroes. I need to ensure ...
0
votes
5answers
55 views

What is the Inverse function of $y = 10^{-x}$? Steps are appreciated.

What is the inverse of $y = 10^{-x}$? These are my steps for the problem. Step 1 $y = 10^{-x}$. Step 2 $x = 10^{-y}$ by inverse substitution. Step 3 $10^y(x) = 1$. Step 4 $10^y = ...
1
vote
1answer
26 views

Proof that $A^{-1}=adj(A)/|A|$

I know that inverse of a matrix is given by $adj(A)/|A|$ but I cannot prove it.Nor did I find the proof in my books.Can you guide me?
0
votes
0answers
29 views

Inverse of sum of matrices

Let $A,B$ be invertible positive definite matrices of the same size. My goal is to efficiently compute $(xA + yB + zI)^{-1}$ for many triplets of positive real numbers $(x,y,z) \in \mathbb{R}^3$. ...