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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Solve a matrix product without computing the inverse

If I have these matrix relationship expressed as a factorization: $$\mathbf{A}=\mathbf{B}\cdot\mathbf{C}$$ where they are $\mathbf{A}\in\{0, 1\}^{m , n}$, $\mathbf{B}\in\{0, 1\}^{m , r}$ and ...
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5answers
39 views

One-to-one function's inverse

I've been trying to solve this question for a while and couldn't find the correct way. We're looking for the inverse of the given function $r$ in terms of $f^{-1}$, where $r$ is defined by: $$r(x) = 1 ...
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20 views

Inverse Fourier transform using laplace

We have to solve the inverse FT of $$\frac{1}{1+4w^2}$$ I tried to do the synthesis but got mediocre results. However this term screams laplace to me. I can see a sine in there. The last lecture they ...
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4answers
30 views

How to verify the inverse of a polynomial in mod polynomial?

This is in $F_2$. This might sound silly but I know that the inverse of $(x^3+x)$ in mod $(x^4+x+1)$ is $(x^3 + x^2)$ but I am not sure how to verify that. It should be that when I multiply $(x^3+x)$ ...
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1answer
18 views

Mathematical function with input not in definition

I have just come across a definition for a mathematical function where in the input is not part of the function definition. This is a simplified variation of the function: $f(x) = \sin(a) + b$ ...
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53 views

Is it possible to inverse a sum of exponents

I have a problem, I need to inverse a sum of exponents. Is it possible? I have this function $y = 0.84826731\times e^{-1.10973369x} + 0.17939312\times e^{-0.1902204x} + 0.02965983\times ...
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0answers
12 views

Series Reversion for $n$ power series

I have $n$ functions with power series representation as $F_i(X)=\sum_{k_1,\dots k_m}a^{i}_{k_1,k_2,\dots k_m}x_1^{k_1}x_2^{k_2}\dots x_n^{k_n}$, where $X=[x_1,x_2,\dots,x_n]$ and ...
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13 views

How can I calculate the inverse fourier transform of $jw$

I am trying to solve $h_I(t)$, which is satisfying $h(t)*h_I(t)=\delta(t)$. e.g. If $h(t) = \delta(t+c)$, then $h_I(t)=\delta(t-c)$. If $h(t) = u(t)$, then $h_I(t)=\delta'(t)$. Q) What is the ...
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1answer
22 views

How do I Inverse Laplace $\frac{(s+1)^3}{s^4}$

I missed a class this week in maths and been a bit lost since with Inverse Laplace, how do I go about finding the Inverse laplace of: $$\frac{(s+1)^3}{s^4}$$ Do I simply expand the numerator? then ...
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22 views

Use conformal mapping to get a heart shape from a square

I use what I call "inverse" conformal mapping in order to properly handle integer locations of output pixels. In other words, if z = x + iy = (x, y) is an output pixel, then I use f(z) = (X + iY) ...
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1answer
35 views

Polynomial modular inverse

I am trying to understand the modular inverse of a polynomial. Let $A , Q$ be polynomials; what is the polynomial $B$ such that $A B = 1 \pmod Q$? I tried searching articles from Finding inverse of ...
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2answers
63 views

Given that $ A$ is a square matrix such that $ A^2 -4A -3I =0$ how do I find $(A+2I)^{-1}$

The way I tried to solve this is to find out how much $(A+2I)$ equals then find it's inverse so: $$A^2 -4A -3I == A^2 -3A-A -2I-I=0$$ $$A^2 -3A-I = (A+2I)$$ Do I simply just inverte the left ...
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1answer
20 views

matrix algebra with invertibles

Im asked to solve for X given the equation $$ (A^{-1}X)^{-1} = (AB^{-1})^{-1}(AB^2) $$ What I have done so far is: $$ ((A^{-1}X)^{-1})^{-1} = ((AB^{-1})^{-1}(AB^2))^{-1} \\A^{-1}X = ...
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4answers
44 views

Inverse function of $f(x) = \frac{x+5}{x-2}$

Find the inverse of the function $f(x) = \frac{x+5}{x-2}$ Here's what I have so far: $y = \frac{x+5}{x-2}$ $x = \frac{y+5}{y-2}$ $(x)(y-2) = (y+5)$ but this seems to be a dead end. How should I ...
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0answers
21 views

inverse square root of band matrix

this is my first post in this web site and I hope that I find an answer to my question. I am trying to find a closed-form expression for the inverse square root of the following symmetric band matrix ...
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3answers
137 views

What is the inverse function of $e^x +x$?

As the natural $\log(x)$ function is the inverse of the exponential $e^x$ and $\log(x +1)$ is the inverse of $e^x - 1$, what it the inverse of $e^x + x$?
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2answers
53 views

what does it mean for the transpose of a matrix to be the negative of the matrix?

Say I have matrix A, if the transpose of A is equal to A then A is symmetrical. but what does it mean if the transpose of A = -A and can I know something about such a matrix inverse?
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1answer
24 views

I'd like to know the inverse of this function:$ y = 100x + 5 (x - 1) (x / 2)$

I'm developing a game with a ranking system and I'm using the formula $$y = 100x + 5 (x - 1) (x / 2)$$ to figure out how much XP is needed to obtain a certain rank. Now the problem is, I also need to ...
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9 views

inverse and unitary invariance

I want to simplify the expression $(\mathbf{V}^H \mathbf{X} \mathbf{V} )^{-1}$, where $V \in \mathbb{C}^{M\times d}, M>d$ is unitary but not square, i.e., $\mathbf{V}^H\mathbf{V}=\mathbf{I}_d$ but ...
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27 views

Consider the relation … What are the domain and range of R? Define the inverse relation. What are its domain and range?

Consider the relation R={(x,y)∈ℝ×ℝ:y=2x}. What are the domain and range of R? Define the inverse relation. What are its domain and range? So I was thinking since there is no set given that it would ...
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2answers
37 views

Inversion of a Block Matrix

Let $S$ to be a symmetric and positive semi-definite matrix of size $n$. What is the inverse of the following block matrix $$ M_{2n\times 2n}= \begin{bmatrix} aI+S & -I\\ -I & aI+S ...
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93 views

How to find the range of $1 / (1+x^2)^{1/2}$?

How to find the range of $$\frac{1}{\sqrt{1+x^2}}$$? Ok. I've revised the (easy theory). I would like to complete the exercise finding the derivative of f(x) and setting equal to zero. I do it ...
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0answers
16 views

Special structured matrices manipulation and inverse

As apart of a bigger analysis I'm doing, I obtained symmetric matrices of a special structure such as $$ \mathbf{A}=\left[\begin{array}{cccc} \alpha_{2}G_{11} & \alpha_{1}G_{12} & \cdots ...
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26 views

On generalised inverse

Let $A$ be a positive matrix, may not be invertible. I define its generalised inverse as \begin{equation} A^- = \lim_{n\rightarrow \infty} \left( \frac{1}{n} I + A\right)^{-1}. \end{equation} Lets ...
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1answer
27 views

How to simplify expressions like $\sinh(4\,\text{arcsinh}(x))$?

I understand that expressions like $\sinh(\text{arcsinh}(x))$ simplify immediately and expressions like $\sinh(\text{arccosh}(x))$ tend to simplify after some algebra. However I cannot work out how ...
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57 views

Find the Inverse Matrix of a Transformation

Let $ f: \mathbb R^3 \rightarrow \mathbb R^3$ be a linear mapping which reflects $\bar{x}$ over the plane $x_1+x_2+x_3 = 0$ . You are given the standard matrix for $f$ is: $$ ...
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17 views

can i compute the inverse mod m by inverting the prime factors?

I have $a \in$ Z/mZ and $a = p_1 * p_2$ in $Z$ (the set of integers, $p_i$ are primes, sorry for not getting the correct format here). Furthermore, it holds $gcd(a,m) = 1$, so there exists an $a^{-1} ...
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75 views

What is the inverse to hyperoperation for positive integers?

According to Wikipedia hyperoperation for positive integers is defined as $$ H_{n}(a,b)=H_{n-1}(a,H_{n}(a,b-1)) $$ with some base conditions. Question: Recursivly define a sequence of binary ...
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2answers
96 views

Given that $(g \circ f)$ is invertible, can we conclude that $f$ and $g$ are invertible?

Given that $(g \circ f)$ is invertible, can we conclude that $f$ and $g$ are invertible? I have previously proved that if $f$ and $g$ are invertible, then $g \circ f$ is invertible, however I am not ...
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1answer
72 views

Is $g \circ f$ invertible if both $f$ and $g$ are invertible? [duplicate]

Is $g \circ f$ invertible if both $f$ and $g$ are invertible?. I know that $f: A \rightarrow B$ and $g: B \rightarrow C$. This is what I have so far: If $f$ is invertible, $f^{-1} : B \rightarrow ...
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36 views

How to solve $n\ln^{2}(\ln 2^{n}) = g(k)$ for $n$?

I've been trying to find the inverse of an asymptotic function for personal research, and I've gotten it down to: $$n\ln^{2}(\ln 2^{n}) = \exp(\frac{9}{64}\ln^{3}(2^{k})))$$ where $\ln n$ is the ...
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26 views

How can I use the Lagrange Inversion Theorem?

I have a function $f(x)= x(\ln(x\ln x))$ and I want to use the Lagrange Inversion theorem to find its inverse $g(x)$ centered around a point $a$. The formula states that: $$g(x)=a+\sum_{n=1}^{\infty} ...
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1answer
41 views

Find value of $g(x)$ where it is the inverse of $f(x)$

Find $g′\left(-\dfrac{1}{9}\right)$, where $g(x)$ is the inverse of $f(x)=\dfrac{x^{17}}{(x^2+8)}$. What is $g\left(-\dfrac{1}{9}\right)$ and $g'\left(-\dfrac{1}{9}\right)$? I tried setting ...
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1answer
51 views

Can the system $x+y=3$, $2x^2 + y^2 = 5$ be solved using matrices?

$$ x+y=3 $$ $$ 2x^2 + y^2 = 5 $$ I solved it by substituting $x = 3- y $ $2(3-y)^2 + y^2 =5 $ therefore, $ y= 2+\frac{i}{\sqrt3} $, $y= 2-\frac{i}{\sqrt3}$ However, I want to know that can I ...
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55 views

Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$.

Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$. Is the converse true? I.e., if $g \circ f$ is invertible, does it follow that $f$ and ...
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5answers
276 views

Is there a difference between $(x)^{\frac{1}{n}} $ and $\sqrt[n]{x}$?

Is there a difference between $(x)^{\frac{1}{n}}$ and $ \sqrt[n]{x}$ ? I'm confused with this topic. Any ideas or examples ? If $(x)^{\frac{1}{n}} = \sqrt[n]{x}$ Consider $x=\frac{-b \pm ...
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81 views

Inverting a matrix in $\mathbb{Z}/n\mathbb{Z}$.

So in my Linear Algebra course I was shown that we cannot directly use row reduction to invert a matrix over a commutative ring in general because the algorithm requires elements to be invertible ...
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33 views

Finding the inverse function of $f(x)=\frac{3x+1}{2-7x}$

Find the inverse function of: $$f(x)=\frac{3x+1}{2-7x}$$ I did the question and when I checked my answer with the key it was wrong, can someone please show me how to properly do this problem? I ...
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0answers
16 views

Inverse of a vector function

How i can prove that this map $f (\theta,\phi)=(\sin (\theta)\cos (\phi), \sin (\theta)\sin (\phi) ,\cos (\theta))$ with $0 <\theta <\pi , 0<\phi <2\pi$, has continuos inverse?
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1answer
37 views

Proof: Inverse of a Matrix

I know that to find the inverse of a matrix, I need to divide 1 by the determinant of the matrix followed by multiplying it by the adjugate of the matrix. However, what is the proof for this? I know ...
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1answer
23 views

Can this diagonal matrix be similar to it's negation?

Suppose I have a diagonal matrix such as $$ A:= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \\ \end{bmatrix}, $$ Is there a ...
3
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2answers
40 views

How to Integrate $1/\left(x\sqrt{1-4x^2}\right)$?

How do I integrate: $$\frac 1t\sqrt{1-4x^2}$$ I am thinking about (Integrals of Inverse Hyperbolic Function): $$\frac{-1}{a} \operatorname{arcsech}\left(\frac xa\right)+C$$ But do I need to use ...
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22 views

Need help proving that a matrix with a specific structure is non-singular

I am working on an engineering problem that requires finding the solution to the following system of linear equations: $$ \underbrace{\left(AB+C\right)}_M\boldsymbol{k}=\boldsymbol{y}$$ where ...
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36 views

Inverse of block triangular matrix

How to find the pseudo-inverse of the following block lower triangular matrix? $$X=\begin{bmatrix} A & 0 \\ c & d \\ \end{bmatrix}$$ Where $A$ is a $n\times n$ lower triangular matrix, $d$ is ...
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1answer
28 views

Confusion with orthogonal matrices

Here are two theorems from my textbook: https://imgur.com/a/UkDUD Why would the projection of y onto W not just be y times the identity matrix? Since an orthogonal matrix times its transpose is the ...
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0answers
90 views

Linear transformation is bijective if it has an inverse

$T : R^{n\times m}\to R^{m\times m}$, $C \to (AC)^T$ I need to find the inverse of this linear transformation, when A is an invertible matrix... Can anyone help me with this problem? I am asked to ...
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1answer
14 views

CDF RV Let Y = X^2. Compute the pdf fY (y). Since X = pY ,

Let $X$ be a continuous random variable with pdf $$f_X(x)=\begin{cases}6x^{-7}&\text{if }x\ge1\\0&\text{otherwise.}\end{cases}$$ Let $Y=X^2$. Compute the pdf $f_Y(y)$. Since $X=\sqrt ...
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0answers
19 views

Properties of frame operator of a matrix

We know many properties of gram matrix $\mathbf{X^TX}$, but what are the properties a frame operator of a matrix i.e., $\mathbf{XX^T}$ and what it tells us about $\mathbf{X}$ ? By properties i mean ...
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1answer
12 views

Expand matrix identity?

What are the intermediate steps to show the following? $$ (I+P)^{−1}(I+P−P) = I−(I+P)^{−1}P $$ I'm looking at the lecture slides here: http://www0.cs.ucl.ac.uk/staff/g.ridgway/mil/mil.pdf
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1answer
26 views

Find $g'(\frac{-1}{2})$ and $g''(\frac{-1}{2})$

Let $f(x)=\frac{x^3}{x^2+1}$, and $g(x)$ is the inverse function of $f(x)$. Then $f(-1)=\frac{-1}{2}$ and $g(\frac{-1}{2})=-1$. Find $g'(\frac{-1}{2})$ and $g''(\frac{-1}{2})$. I have found ...