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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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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1answer
59 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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2answers
39 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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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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3answers
97 views

Inverse of a matrix with $a+1$ on the diagonal and $a$ in other places

Let $a>0$. Let $A$ be the $n\times n$ matrix with $a+1$ on the diagonal and $a$ in all other entries. How can one compute $A^{-1}$ as a function of $n$?
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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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2answers
757 views

Inverse functions and tangent line

Let $f(x) = \frac14x^3 + 12x + 6$ and let $y = f^{-1}(x)$ be the inverse function of $f$. Determine the $x$-coordinates of the two points on the graph of the inverse function where the tangent line is ...
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0answers
12 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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1answer
101 views

complex and decimal tetration

So recently in the blog post on tetration, it talked about tetration with nice clean powers (calling them these because I don't know the right term). But how does it work when given a complex power? ...
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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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4answers
668 views

is it true every left inverse of a matrix is also right inverse of it?

I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ ...
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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
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
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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1answer
46 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
745 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
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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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4answers
19k views

Inverse function of a polynomial

What is the inverse function of $f(x) = x^5 + 2x^3 + x - 1?$ I have no idea how to find the inverse of a polynomial, so I would greatly appreciate it if someone could show me the steps to solving this ...
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2answers
46 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
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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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
27 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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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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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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2answers
189 views

Is the pseudoinverse of a singular, lower triangular matrix itself lower triangular?

Suppose $L\in\mathbb{R}^{n\times n}$ is a singular, lower triangular matrix. Is its psuedoinverse, $L^\dagger\in\mathbb{R}^{n\times n}$, also lower triangular? I have already proved by induction that ...
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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
108 views

If $f:\mathbb R\to\mathbb R$ continuous does $f^{-1}$ also continuous?

Let $f:\mathbb R\to\mathbb R$ is bijective and continuous, does $f^{-1}$ is also continuous ? Does this result hold for $f:U\to\mathbb R$ where $U\subset \mathbb R^n$ ? and for $f:V\to\mathbb R^m$ ...
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1answer
1k views

A continuous bijection $f:\mathbb{R}\to \mathbb{R}$ is an homeomorphism?

A continuous bijection $f:\mathbb{R}\to \mathbb{R}$ is an homeomorphism. With the usual metric structure. I always heard that this fact is true, but anyone shows to me a proof, and I can't prove it. ...
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6answers
9k views

Finding the inverse of $h(x) = 3^x$

most of the time I know how to find the inverse of a function (make it equal $y$, solve for $x$ and then swap $x$ and $y$), but I have no idea how to do that for this one, so any help would be great: $...
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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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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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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
37 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 ...
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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 ...
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2answers
379 views

Inverse of the Modified Bessel function

Is there any chance of having a formula or approximation to inverse the Modified Bessel function of the first kind? I mean to solve $I_M(x)$ with respect to x: $I^{-1}_M(x)$? Thanks in advance
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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 $B$, $A$ ...
3
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2answers
493 views

How would I show this bijection and also calculate its inverse of the function?

I want to show that $f(x)$ is bijective and calculate it's inverse. Let $$f : \mathbf{R} \to \mathbf{R} $$ be defined by $f (x) = \frac{3x}{5} + 7$ I understand that a bijection must be ...
2
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1answer
2k views

Trick: Substitution in inverse trigonometry.

My friends say, it is some what difficult to know, which trigonometric function has to be substituted in the inverse trigonometric equations, to get the correct solution. So, I thought to take up this ...
2
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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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1answer
2k views

Find an inverse of $a$ modulo $m$ for each of these pairs of relatively prime integers

How would I find the inverse of a given number $a$ modulo $m$, given that $\gcd(a,m)=1$? a) $a = 2$, $m = 17$ $17 = 2 \cdot 8 + 1$ $2 = 1 \cdot 2 + 0$ $1 = 17 - 8 \cdot 2$ <-How do I know ...
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0answers
86 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:...
0
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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 ...
0
votes
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 ...