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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24 views

Inverse modulo certain numbers.

Let $b,n$ be given positive integers. I would like to give some formula for the multiplicative inverse $a^*$ of a positive integer $a$ to a modulo of the form $4ab-n$, where $(a,n)=1$, i.e. $$ aa^*\...
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1answer
33 views

Row sum of inverse of a matrix

Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix} $$ All the elements of A are ...
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1answer
31 views

Right inverse matrix

I know that if $A, B$ and $C$ are square matrices such that $$ AC=I \quad \mbox{and} \quad BA=I, $$ then \begin{eqnarray*} AC=I & \Rightarrow & BAC=B\\ & \Rightarrow &IC=B\\ & \...
2
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1answer
58 views

How can I divide a vector by a matrix?

I am trying to go backwards through a neural network. I have an output and I want to see what input would lead to that output. To go forwards I start with a vector and multiply by a matrix and then ...
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1answer
19 views

Inverse Parameters of a Pan-Tilt Rotation Possible?

I have a 2-parameter (tilt,pan) rotation computed as tilt followed by pan, i.e. two rotation matrices multiplied together: $$R(t,p)=\begin{pmatrix} c_p & s_p s_t & s_p c_t \\ 0 & c_t &...
2
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1answer
184 views

is this matrix invertible

Is the following matrix invertible? $\left[ \begin{matrix} \sum\limits_{x=1}^{n}{1} & \sum\limits_{x=1}^{n}{x} & \sum\limits_{x=1}^{n}{{{x}^{2}}} & \cdots & \sum\limits_{x=1}^{n}{{...
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5answers
48 views

Inverse Equation of the Given Equation

Having a bit of a problem getting the inverse of the following equation: $$f(x) = \sqrt{9-x^2}$$ I had an answer which was equal to $3-x$ but when I used sites like Mathway and Wolfram to check my ...
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0answers
24 views

Linear regression of matrix elements to get the minimal polynomial to perform a matrix inversion?

So each matrix $\bf A$ fulfils an equation for it's minimal polynomial $P_m({\bf A})$: $$P_m({\bf A}) = 0 \Leftrightarrow \sum_{k=0}^{k_n}c_k{\bf A}^k = 0$$ We can by multiplying with $A^{-1}$ and ...
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3answers
42 views

Calculate multiplicative inverse of $95$ in group of order $n=101$ which is subgroup of $(\mathbb{F}_{607}^*,\cdot)$

In the notes where I'm studying from there is written: "Let $G=\langle g\rangle$ be a subgroup of $(\mathbb{F}_{607}^*,\cdot)$ with $g=64$ and order $n=101$" but that felt strange to me; since I know ...
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0answers
24 views

Solving for $f_2(x, y)$ in $\int_{\mathbb{R}}f_1(x, y) f_2(x, y) dx = g(y)$

Let me first present the problem in its most general form, and then work down to a specific case of interest and my motivation. Suppose $f_1: \mathbb{R}^2 \rightarrow \mathbb{R}_{\ge 0}$ and $g: \...
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1answer
13 views

Square Root of a matrix: transpose or inverse of eigen vectors?

Here is described that the square root of a matrix is defined as K^1/2 = V*D^1/2*V^-1 At the end of scetion 4 of this paper we can see W = K^-1/2es In the ...
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1answer
51 views

How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix?

How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix? Assume that $M$ is an $n \times m$ matrix with $n \leq m$. Is it possible to obtain ...
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0answers
9 views

Is there a simple algorithm to compute polynomial inverses over cyclotomic polynomials?

I'm working with polynomial inversions in a ring built over the nth-cyclotomic polynomial, with $n = 2^i$. As usual, I'm applying Extended Euclidean algorithm on this, an approach that does not scales ...
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1answer
40 views

Is this how eigenvalues of some matrix $A$ are related to the inverse of $A$?

Let $A$ be an invertible $n\times n$ matrix. If $$Av = \lambda v \qquad (1)$$ for some $v$ and $\lambda$ then $\lambda$ is an eigenvalue of $A$ and $v$ a corresponding eigenvector. Equation $(1)$ may ...
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1answer
50 views

Inverse operation of tetration and how it is computed?

If $c=a+b$, then $a=c-b$ and $b=c-a$. If $c=a\times b$, then $a=\frac{c}{b}$ and $b=\frac{c}{a}$. If $c=a^b$, then $a = \sqrt [b]{c} =c^{\frac{1}{b}}$ and $b=log_ac$. What are the analogous inverse ...
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2answers
41 views

Is $\text{arccosec}(x) = \arcsin\left(\frac{1}{x}\right)$ for all $x \in ℝ?$

Is $\text{arccosec}(x) = \arcsin\left(\frac{1}{x}\right)$ for all $x \in ℝ?$ I'm still really new to trigonometric inverses, so if the above was cleared up I'd be grateful. Thanks.
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3answers
32 views

Complex inverse function

I've got a problem when solving an inverse function. Usually when I have a basic function and trying to find its inverse is not a problem. I just solve for X and find it. But now I've got a more ...
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1answer
57 views

Prove a matrix expression leads to an invertible matrix?

I want to prove matrix $C$ is invertible: $$C=I-A^TB(B^TB)^{-1}B^TA(A^TA)^{-1},$$ where $I$ is an identity matrix of appropriate dimensions, and $(A^TA)^{-1}$ and $(B^TB)^{-1}$ imply both $A$ and $B$ ...
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1answer
56 views

How to show the matrix $\left( \binom{x-i}{j-1}\right)_{1\leq i,j\leq 2r+1}$ has determinant (-1)^r and it's inverse?

After playing around in mathematica, I found that the matrix $\left( \binom{x-i}{j-1}\right)_{1\leq i,j\leq 2r+1}$ has determinant $(-1)^r$ for the first few $r$'s. How can I prove this this, or at ...
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2answers
44 views

Uniqueness of Inverse

I am having trouble understanding the logic for a few steps in the following. I'll point the steps out at the end. If B and C are inverses of a square matrix A, then B = C. Proof: Since B is an ...
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2answers
37 views

Finding the expression of the inverse of $(AB)^T$

I know that $(AB)^T$ = $B^TA^T$ and that $(A^T)^{-1}= (A^{-1})^T$ but couldn't reach any convincing answer. Can someone demonstrate the expression.
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1answer
44 views

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

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?
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0answers
42 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
37 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
44 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
67 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
34 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
38 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
16 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
63 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
57 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
51 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
35 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
23 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
56 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
48 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
39 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
23 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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1answer
22 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 $$...
0
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4answers
50 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 ...
1
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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 $y=x$ ...
0
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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
41 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
32 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 ...