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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Inverse of the sum of two orthogonal projections

I am trying to find out, if there is a formula for finding the inverse of the sum of two orthogonal projections. So basically my questions is: If $\left[\mathbf{A},\mathbf{B}\right]$ is full rank, ...
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57 views

Suppose R is an integral domain. Prove that $(a)=(b)$ if and only if $b = ua$ where $u$ is in $R^\times$

I am lost on this one. I'm still new to ring theory, as we're only a couple weeks into the course, but it's already well over my head. I know that $R$ is an integral domain, so the additive and ...
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47 views

Inverse of $f(x)=\sin(x)+x$

What is the inverse of $$f(x)=\sin(x)+x.$$ I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet. What about $$f(x)=\sin(a \cdot x)+x$$ where ...
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1answer
42 views

Expressing an inverse trig function?

I just need a little help with this question: "Express cos$y$ in terms of cos $y/2$ and hence show that tan$^{-1} sqrt[(1-x)/(1+x)] = 1/2$ cos$^{-1}x$, for $0<x<1$." I can do the first part, ...
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1answer
20 views

Derivation of inner variations

In Giaquinta's and Hildebrandt's 1996, "Calculus of Variations 1", pages 147-148, they develop the definition of inner variations. They first fix $\lambda\in ...
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44 views

inverse of quadratic matrix form

I have an expression of the form: $ACA′$ where C is an invertible, symmetric and positive definite matrix. I'm trying to figure out if the expression above is invertible (or what additional ...
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1answer
115 views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
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71 views

Is there a name for an algebraic structure like this?

I'm self studying abstract algebra. I see that in rings there's no requirement for a multiplicative inverse. Is there something similar except with no requirement for an additive inverse. For ...
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46 views

Uniform continuity of inverse in only one variable

Let $f:[0,1]\times[0,1]\to \mathbb{R}$ be a (uniformly) continuous functions. Denote the image of $f$ by $D_f:=\{(x,y): x\in[0,1] , 0\leq y \leq f(x,1)\}$ $f$ is such that the section $f_x$, i.e. the ...
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63 views

Increase the diagonal entries of a positive definite matrix

Assume that we have a positive definite matrix $C$, and a positive definite diagonal matrix $\Lambda$. Are all the diagonal entries of $(C + \Lambda)^{-1}$ smaller than those of $C^{-1}$? In other ...
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31 views

Matrix problem involving an equation.

Please could you help me with the below question. There are three parts, and all of my working is displayed! Thank you in advance, kind stranger. For an integer n, real numbers a,b,c and an nxn ...
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29 views

Error bounds in representing a vector using a truncated Moore-Penrose biorthogonal basis

I was reading and trying to reproduce the results in the arXiv preprint of Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression by Asaf ...
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8answers
76 views

Find the inverse of the following matrix.

How can I calculate the inverse of $M$ such that: $M \in M_{2n}(\mathbb{C})$ and $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$, and I find that $\det M = 2^n$. I tried to find the ...
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2answers
74 views

Why is $3^n$ not in $\Theta(2^n)$

How is it that $3^n$ not in $\Theta(2^n)$, while $log_3 n$ is in $\Theta(log_2 n)$ ?
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31 views

Pseudo-inverse of an underdetermined Toeplitz matrix

I have an undetermined Toeplitz matrix (more columns than rows). For example: \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 ...
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2answers
32 views

how to find inverse laplace transform of

how to find the inverse laplace transform of $\frac{s}{s^4+s^2+1}$. I tried to do it via partial fraction and reached $\frac{s}{(s^2-s+1)(s^2+s+1)}$
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45 views

Inverse of $f(x) = 18sin(\frac{x\pi}{7})+20$

This is an exercise taken from Mooculus-textbook (page 17, exercise 5 to be exact). The task given is to find an inverse for $f(x) = 18\sin(\frac{x\pi}{7})+20$ (restricting domain to $[3.5,10.5]$) ...
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109 views

calculator issue: radians or degrees for inverse trig

It's a simple question but I am a little confused. The value of $cos^{-1} (-0.5)$ , is it 2.0943 or 120 ?
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3answers
115 views

Do continuous mappings always have an inverse?

There's a theorem that states that a mapping $f$ from $X$ to $Y$ is continuous if and only if the inverse image of any open set in $Y$ is open in $X$. Does this mean that continuous functions always ...
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73 views

Does a smooth mapping always have an inverse map which is also smooth?

If not, can someone provide counterexamples? Thank you
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47 views

Optimal series expansion for “invertability”

Motivation: Often when dealing with physical phenomena, deviations from the model must be considered, so a variable, say $x\in[0,1]$ will be replaced by a power series expansion: $$x'\ \to \ x(1+k ...
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119 views

Invertible function $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $

How can I prove that $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $ is invertible. First I choose variable $x$ for $y$ and tried to switch and simplified the function but I am stuck. Need some help ...
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41 views

Integrating inverse functions

I'm trying to integrate the following: $$\int_0^1 \left[\frac{c}{(1+c^{-1}(\tilde{b}))}\right]dc$$ If it helps ...
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1answer
36 views

Invertibility of a Matrix Given Some Conditions

Let $A$ and $B$ be different $n\times n$ matrices with real entries. Suppose that $A^3=B^3$ and $A^2B=B^2A$, can $A^2+B^2$ be invertible?
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62 views

Inverse Z transformation in specific points

I'm given $$H(z)=\frac{z^4+6z}{z^6+1}$$ and I need to find $h(k)$ for $k=0,1,2,3,4$. Where $H(z)$ is the Z transformation of $h(k)$. Since H is very complicated, I believe some trick could be ...
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34 views

Inverse Z transformation of 1/(z^2(z^2+1)^2)

I need to find the inverse Z transformation of $\frac{1}{z^2(z^2+1)^2}$ So far, I've tried using the convolution property, and so, inverting $\frac{1}{z(z^2+1)}$, gave me $-0.5i({{(-i)}^k-i^k})$ for ...
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2answers
26 views

Matrix inversion with variable in {-1,1}

Could you please give me a hint for computing inversion of this matrix? $$ \begin{pmatrix} 1 & f & g+h\sqrt(2) \\ 0 & i & j \\ 0 & 0 & 1 \\ ...
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1answer
209 views

Inverse of binary matrix

I have tried creating an inverse of a binary matrix using the identity matrix method. Have an identity matrix alongside the square matrix and perform all the operations to convert the square matrix to ...
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1answer
33 views

Solving: How to find an inverse function for this function?

I got this example: and I am trying to find an inverse function to this function. Could I ask you, please, how to do that? Thank you
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1answer
67 views

Reversion of power series

So, I just heard about this method. How does one determine the coefficients, and what is it used for? For example, given $$ y = x - \frac{x^3}{6} + \frac{x^5}{120} + O(x^7)$$ reversion would give a ...
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32 views

Maximal value of domain for a function by looking at inverse function.

The function g:[–a,a]→ R, g(x)=sin(2(x-π/6))has an inverse function.The maximum possible value of a is: From what I understand the domain of g(x) is the range of g'(x). So I would try to find the ...
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278 views

Calculating Moore-Penrose pseudo inverse

I have a problem with a project requiring me to calculate the Moore-Penrose pseudo inverse. I've also posted about this on StackOverflow, where you can see my progress. From what I understand from ...
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1answer
34 views

Trouble with derivation involving Inversion of partitioned Matrix

$$\alpha=Q'\beta=\begin{pmatrix}\alpha_1\\\vdots\\ \alpha_p\end{pmatrix}, f=\begin{pmatrix}\delta_1\alpha_1\\\vdots\\\delta_p\alpha_p\end{pmatrix},F=\begin{pmatrix}\delta1\alpha_1& & \\ ...
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603 views

How to calculate the inverse of a complex matrix?

How can I calculate the inverse of $$H = \pmatrix{ h_{00} & h_{01} \\ h_{10} & h_{11}},$$ where $h_{00}$, $h_{01}$, $h_{10}$, and $h_{11}$ are complex numbers?
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69 views

Inverse of a sum of PSD matrices

I was wondering if anyone knew any techniques to convert the following: $ (A+B+C+..)^{-1} $ where $A,B,C...$ are positive semi-definite (PSD) matrices into a sum of some other function: $ ...
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1answer
32 views

Invertability of Tensor Product of a Square Positive Definite Vandermonde Matrix with itself

Given the tensor product of a Invertable Square Positive-Definite Vandermonde Matrix $a$ $$\mathbf{a} = \ \left( \begin{array}{ccc} 1 & 1 & \ldots & 1^{D-1} \\ 1 & 2 & \ldots ...
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95 views

Inverse of Ulam's spiral

I have a program and I need a function that takes a coordinate as input and returns an integer corresponding to the position in Ulam's spiral. The simple (but slow) way to do this would be to ...
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42 views

two short doubts about the inverse function in a point

the function is $F(x,y,z)=(y^2+z^2, z^2+x^2, x^2+y^2)$ the point is (-1,1,-1) task: find the local inverse of F in that point. I have already proved that F is actually invertible there. then i ...
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89 views

Inverse function of $y=2x+\sin x$

I was doing a long exercise when come to this point: calculate the inverse function of $y=2x+\sin x (x \in\mathbb R) $ and its derivative. I know that the derivative of an inverse function is ...
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2answers
115 views

Inverse modulo question?

I know that when gcd(a,b) = 1, a and b are relatively prime. This allows you to write the linear combination aS + bT = 1, where S and T are Bezouts's coefficients. As I understand, one of these ...
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1answer
34 views

Calculate the inverse of a complex matrix

I am trying to calculate the inverse of a given matrix but somewhere I have an error in my calculation that I cannot find $$\begin{array}{ccc} && \left( \begin{array}{ccc|ccc} 1-i & 2 ...
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23 views

probability subspaces that make entropy function equal to a constant value

Given the entropy fucntion: $$ H = -\sum_i^n p(i) \ln(p(i))\,.$$ where $p(i)$ are probabilities and $n=4$, I would like to know all the points in the probability space that make $H = k$, being $k$ a ...
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44 views

Taking the (pseudo)inverse of a monoid operation.

Let $M$ be a monoid with binary operation $f : M \times M \to M$. I'm interested in functions $g : M \to M\times M$ that obey the property: $$ f(g(m)) = m $$ I want to understand what all of the ...
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Are most matrices invertible? [duplicate]

I asked myself this question, to which I think the answer is "yes". One reason would be that an invertible matrix has infinitely many options for its determinant (except $0$), whereas a non-invertible ...
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Given the inverse of a block matrix - Complete problem

Given $X$ a block matrix $$\pmatrix{A&B}$$ where $A$ is $m \times n$ and $B$ is $m \times (n−m)$. I know a priori the value of $X \times (X^{T} \times X)^{-1}$. Substituting $X$: ...
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175 views

A problem with the geometric series and matrices?

Let $n$ be a positive integer. Let $A$ be a square matrix. Let $I$ be the identity matrix with the same size as $A$. I want to simplify $f_n(A) = I + A + A^2 + A^3 + A^4 + \cdots + A^n$ Now I know ...
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104 views

If matrix A is invertible, is it diagonalizable as well?

If a matrix A is invertible, then it is diagonalizable. Is it true or false?
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67 views

Given the inverse of a block matrix…

Given the inverse of a block matrix $X^{-1}$, where $$ X=\left(\begin{array}{cc} A & B \end{array}\right). $$ A is $m\times n$ and B is $m\times(n-m)$. Can I obtain the pseudo-inverse of A ...
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1answer
33 views

Function composition and inverse

Consider f : ℝ \ {1} → ℝ \ {1} given by f(x) = x/(x-1) I need to find: 1) f ◦ f ◦ f and 2) the inverse function f^-1(x) So far I have: 1) f(f(x/(x-1)) = f(x) = x/(x-1) which is suspicious to me ...
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25 views

Calculate inverse modulo: $8^{-13}\pmod {29}$

How can I calculate $8^{-13}\pmod{29}$ ? I don't get how it works. Can I do it separately? So first $8^{-13}$ and then modulo $29$. And how can I calculate a negative power the quickest?