Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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33 views

What are all the Banach algebras where $\|a\|\|a^{-1}\|=1$? [on hold]

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?
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0answers
23 views

$y=x/(1+a(x))$, $\quad$ $x=y/(1+b(y))$. What is known about $a\mapsto b$?

\begin{align} y & = f(x) = \frac x {\displaystyle 1 + \sum_{n=1}^\infty a_n \frac{x^n}{n!}} \\[15pt] x & = f^{-1}(y) = \frac y {\displaystyle 1 + \sum_{n=1}^\infty b_n \frac{y^n}{n!}} \end{...
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1answer
36 views

Kernel of inverse of the Möbius transformation

Given $f(z)=\frac{az+b}{cz+d}$ the Möbius transformation. Calculate $ker(f^{-1}(id))$. $$f(z)=\frac{az+b}{cz+d}=w\implies w(cz+d)=az+b\implies z(wc-a)+wd-b=0$$$$\implies z=\frac{-wd+b}{wc-a}=f^{-1}(z)...
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2answers
25 views

How to normalize and inverse a vector so it sums to 1 ?

I understand how normalization works. You sum up the individual values of the vector, you divide each value by the sum, and voila... they sum to 1. Why doesn't it work when you subtract them from ...
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2answers
42 views

Repeated use of Woodbury formula

I want to calculate the $x$ dependency of $\left(I + A \Lambda (x) A^{T}+B\Omega(x)B^{T}\right)^{-1}$ explicitly, where $I$ is a $n\times n$ matrix. Here $\Lambda (x) $ and $\Omega(x)$ are diagonal $...
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1answer
60 views

Calculating inverse function with 2 variables

$f: R^{2}\mapsto R^{2}$ $(x,y)\mapsto (x^{2}-4y^{2}+x, -xy+3y)$ I should calculate inverse function of $f$ in point $(3,1)$. I tried to do $(x,y)\mapsto(u,v)$, but I just dont know how to get x ...
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0answers
33 views

Inverse of the sum of identiy matrix and a symmetric matrix

Is there a simple way to solve $(I + A) X = B$, where $I$ is the identity matrix, and $A$ is a symmetric matrix?
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1answer
36 views

Inverse of a matrix with main diagonal elements approaching infinity

Let $A$ be a invertible, symmetric, positive definite $p \times p$ covariance matrix with main diagonal elements $a_{ii},~i = 1,~\ldots,~p$. If all main diagonal elements would approach $\infty$, ...
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0answers
13 views

How to solve a LMI with inverse matrix and quadratic form

I have to solve the following LMI, where $\Sigma$ is a symmetric positive definite matrix. K,D and $\Sigma$ are unknown: $$\left[\begin{array}{cc} K\Sigma^{-1}K^{T}+DVD^{T}+I & KA^{T}\\ AK^{T} &...
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3answers
56 views

Inverse of $f(x) = 2x^2+8x+13?$

How can you find the inverse of $f(x) = 2x^2+8x+13?$ This is what I've tried so far: $y = 2x^2+8x+13$ $x = 2y^2+8y+13$ $x-13 = 2y^2+8y$ $x-13=y(y+8)$ This is where I got stuck. To be clear, I want ...
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0answers
50 views

Time complexity of inverting an $n \times n$ matrix which is the sum of a rank-$m$ matrix and a full-rank diagonal matrix

I want to know the time complexity of inverting $K$, where $K$ is an positive-definite $n\times n$ matrix: $$K=\Lambda+Q$$, where $\Lambda$ and $Q$ are both $n\times n$ matrix, $\Lambda$ is a full-...
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1answer
47 views

How could I search the inverse operator $(Af)^{-1}(x)$

I am try to search $A^{-1}$ when I define $A:L^2[0,2] \rightarrow L^2 [0,2] $ when $$(Af)(x)=x^{-1/4}f (\sqrt {2x}) $$ What I do: I consider that $ (Af)^{-1}((Af)(x))=Ix=x \Longleftrightarrow (Af)^{...
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1answer
34 views

Show there exists $C\in\Bbb{R}^n$ such that $|C-A_i|=|B-A_i|+u_i$, with $A_i,B\in \Bbb{R}^n$ and $u_i$ close enough to $0$

Let $A_1,...,A_n,B$ be vectors in the $n$-dimensional Euclidean Space, such that they are never on the same affine $(n-1)$-dimensional subspace. (What? Is that a way to say they span $\Bbb{R}^n$?). ...
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0answers
16 views

Conditional Mean Given Precision Matrix While Avoiding Inversions

I'm working on a problem where I need to compute a conditional mean directly from a precision matrix (the inverse of covariance matrix). Let $\boldsymbol \mu$ be a mean vector partitioned into $$\...
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0answers
21 views

Inverse projection matrix 2D to 3D

I am writing a simple computer vision application in which reports the position of coloured dots on the floor. The floor is observed by a camera for which I have the correct projection matrix. I.E. If ...
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0answers
54 views

Use the complex definition of $\sin z$ to find an expression for $\sin^{-1} z$

Using $$\sin z = \frac{e^{iz}-e^{-iz}}{2i}$$ Prove $$\sin^{-1} z =\frac{1}{i}\ln(iz+\sqrt{1-z^2}) $$ Attempted solution: Let $\sin z = u$ and $e^{iz} = v$. \begin{align*}& 2iu = v - \frac{1}{v}...
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2answers
47 views

Is $ f \circ g $ invertible in the diagram below?

I was working through Can the composition of two non-invertible functions be invertible? For the image below is $f \circ g$ invertible? Thanks!
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0answers
38 views

inverse function of integral and bilateral filter

There is a formula in the bilateral filter thesis $$ h(x)=k^{-1}(x)\int_{-\infty}^\infty\int_{-\infty}^\infty f(ξ)c(ξ,x)s(f(ξ),f(x))dξ\tag{1} $$ $$ k(x)=\int_{-\infty}^\infty\int_{-\infty}^\infty c(ξ,...
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3answers
28 views

Find the inverse of $f(x) = 1 + \frac{1}{x}, x \gt 0$

I'm tasked to find the inverse of the function $$f(x) = 1 + \frac{1}{x}, x \gt 0$$ The book offers a solution, simply to set $$1 + \frac{1}{x} = s$$ and solve $$x = \frac{1}{s-1}$$ and I think I ...
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1answer
22 views

Multidimensional Newton's Method: Inverse Jacobian

How calculate programs/packages like Matlab, Python/scipy, ...the inverse jacobian for multidimensional Newton's method? $x_{n+1} = x_n -(J(x_n)^{-1}*f(x_n)$ How can the Jacobian be calculated? How ...
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4answers
30 views

value of an inverse trignometric expression

How can we find the value of $ 3\sin(\frac12\arccos\frac19)+ 4\cos(\frac12\arccos\frac18)$ ? Substituting A = $\arccos\frac19$ My approach to this question.. I tried to use the formula $\cos A = \...
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2answers
36 views

Inverse image of a function in multivariable calculus?

Let $f: R^2 \rightarrow R^2 $ defined by $f(x,y) = (x+y,xy).$ Claim : Inverse image of each point in $R^2$ under f has at most two elements. My Claim : Suppose $f(x,y) = (x+y,xy)= (p,q).$ We have ...
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3answers
38 views

value of an Inverse trigonometry expression

How can i find the value of $\alpha = \arcsin\frac{\sqrt{63}}{8}$ to substitute in the expression , to value of $\sin^2(\frac\alpha4)$ ?
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0answers
32 views

Inverse Laplace transform of $(s^2-1)^{-1/2}$

please help with this. Not derived from any differential equation. Also found the answer $\mathcal{L}^{-1}(\dfrac{1}{\sqrt{s^2-1}})$
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1answer
19 views

Convergence of the inverse in Sobolev spaces

Assume we have a sequence $f_k$ which converges to $f$ in the Sobolev space $H^p(D)$, where $D\subset\mathbb{R}^N$ ($N\geq 2$) is relatively compact and $p\geq 1$ is an integer. We also assume that $$...
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4answers
44 views

How to find an inverse of the following function?

$$f(x)=x^3+1$$ To find inverse, from what I've learned we change the y to x $$x=y^3+1$$ solve for y $$x-1=y^3$$ Should I cube root the x-1 for this? if i did that I still would not get the answer ...
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1answer
38 views

Find the inverse function of $y=x|x|e^x$

I am having problems finding the inverse function of a complicated function. In this case: $$y=x|x|e^x $$ I thought I could 'split' this function but I'm not sure if that's the right way. for $y=x$ ...
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1answer
17 views

Inverse of bounded linear transformation

I'm not in the mathematics field and not very comfortable with strict mathematical formalism. The information I find on the Internet includes so many technical terms that might take ages for me to ...
3
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0answers
40 views

Additive basis of order n: Sets which allow every integer to be expressed as the sum of at most n members of that set. [closed]

Every integer can be expressed as the sum of at most 3 triangular numbers. That is, the set of triangular numbers is an additive basis of order 3. The sum of the inverse triangular numbers is 2. (1/1 +...
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2answers
31 views

Higher derivatives of inverse functions (Multivariable Calculus)

Given the function $$ (u,v) = f(x,y) = (x + y, x^2 - y^2) $$ I would like to compute the second partial derivative of $x$ with respect to $v$, at the point $(u,v) = (2,0)$. To calculate the ...
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0answers
15 views

Inverse of a special matrix: controlabillability like matrix from control theory

Is there a way to find the first vector in the inverse of the following real matrix $$ M = \begin{bmatrix}B^{T} \\ B^{T} A^{-1} \\ \vdots \\B^{T}A^{-(n-1)} \end{bmatrix}$$ as a function of $B$, $A$ ...
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1answer
56 views

What is the value of $\cos(\tan^{-1}(\tan 2))$?

What is the value of $\cos(\tan^{-1}(\tan 2))$? Am I thinking correct? $\tan 2$ is negative so $\tan^{-1}$ and $-\tan 2$ cancel each other giving $\cos(-2)$ which finally gives the answer as $-\cos ...
0
votes
1answer
55 views

What is the inverse Laplace transform of $\lfloor s \rfloor$?

How can we find the inverse Laplace transform of: $[x]$ (floor function) ? My question isn't LLaplace transform of floor function i asked the "inverse" laplace transform of floor function $$\mathcal{...
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1answer
24 views

Continuous dependence of matrix elements

I've stumbled upon several solution of linear algebra problems which use notion of "continuous dependence" of matrix polynomials on matrix elements. For instance (translated, so any inaccuracies are ...
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0answers
87 views

Math notation to show two numbers in a range that added together get the max of the range [closed]

I am completely new to math notations, it's been about 30 years since high school, and I am writing a research paper (completely on my own, not for a degree). I basically want to show that two real ...
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0answers
39 views

Alternative view of matrix inversion (explanation required)

We were taught in linear algebra that in order to try to find the inverse of a matrix we can create an augmented matrix $[AI]$ where $A$ is the original matrix and $I$ is the identity matrix. Then we ...
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1answer
21 views

Relation between powers of inverse modulo n.

Recently, I was studying enchanced euclidean algorithm. I am wondering if there is some way to calculate inverse of $a^2$ (and higher powers) modulo $n$, knowing inverse of $a$ modulo $n$. For example:...
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3answers
43 views

Value of the given expression …

If $$y=\tan^{-1}\left(\sqrt{\dfrac{1+\cos x}{1-\cos x}}\right)$$ then value of $(2x+14y)^3-343$ is ? I reduced the equation as $y=\tan^{-1}\left(\dfrac{1+\cos x}{\sin x}\right)$ but I couldn't ...
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1answer
28 views

Using inverse of transpose matrix to cancel out terms?

I am trying to solve the matrix equation $A = B^TC$ for $C$, where $A$, $B$, and $C$ are all non-square matrices. I know that I need to utilize $M^TM$ in order to take the inverse. I'm just not sure ...
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1answer
52 views

Differentiation Involving Determinant.

I have to compute the following differentiation : $$\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times n}'...
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1answer
29 views

If a linear eq. System has only a single solution, the matrix has always an inverse? [closed]

If i have a 3x3 matrix of a linear eq. System and i archieve an upper triangular matrix (means unique solution) does this form always have inverse matrix? Thank you very much!
0
votes
1answer
118 views

If $f^{-1}(x)=\frac{1}{f(x)}$ then find $f(1)$

For $a>1$ we have: $f:[\frac{1}{a},a]\to [\frac{1}{a},a]$ be a bijective function. Suppose $f^{-1}(x)=\frac{1}{f(x)}$ for all $x \in [\frac{1}{a},a]$ then find $f(1)$. Could someone give me ...
1
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0answers
36 views

discrete random variable with uniformely distributed random variable

I hope you can help me because I have no clue where to start: Let X be a discrete random variable with $ p_k=P_X[X=x_k]=p(x_k) $for all $1\le k\le N$ for $N\in \Bbb N$ and distribuition function: $$...
2
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1answer
63 views

Function inversion (analytical)

Can $t(x)$ be found from: $$A \, t + B\ln\frac{1-t}{t}=x \; ?$$ Here, $A>0, \; B < 0$ and $0 \lt t \lt 1$. The $t(x)$ should be given in analytical form (even if you use, say, Lambert's W - ...
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1answer
22 views

Using the Affine cipher, do we need $a^{-1}$ if we know gcd(a,26)=1?

I have just attempted the affine cipher with the word "code" $CODE = 02140304$ Lets choose our key as $(5,3)$, so our encryption is $y=5x+3$ $13211823=NVSX$ Now, to undo the code, I would have to ...
5
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1answer
82 views

Can $A^{T}(AA^{T})^{-1}A$ be simplified?

Let $A$ is an $m\times n$ ($m<n$) real matrix with full positive entries and $\text{Rank}(A)=m$. Thus $(AA^{T})^{-1}$ is an $m\times m$ symmetric $M$-matrix since $AA^{T}$ is nonnegtive and ...
4
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1answer
61 views

Inverting an $n \times n$ matrix using determinant

We're asked to invert the following matrix with the help of guided questions. $$\begin{pmatrix} 1 + a_1 & 1 & \cdots & 1 \\ 1 & 1+a_2 & \ddots & \vdots \\ \vdots & \ddots &...
3
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2answers
95 views

Why can't the inverse of $F(x)= x+\sin(x)$ have a formula algebraically?

I'm only curious why the inverse of $f(x)$ can not be determined algebraically. Is it because the inverse of $\sin(x)$ cannot be converted into a formula?
0
votes
1answer
41 views

Finding the inverse Laplace transform of this function

Find the inverse Laplace transform of this function (related to my question earlier): $$f(t)=\mathcal{L}_s^{-1}\left[\frac{s}{s+\frac{1}{\tau}}\cdot\frac{A}{s}\left(1-\mathrm e^{-\frac{Ts}{2}}\right)^...
4
votes
2answers
53 views

When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$

Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$ My attempt: I took A = $$ \begin{bmatrix} 2 & 0 &0\\ ...