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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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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37 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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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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4answers
992 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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159 views

Do continuous mappings always have an inverse?

A theorem of general topology 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 ...
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135 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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52 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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3answers
135 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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46 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
37 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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70 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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31 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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501 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
41 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
210 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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1answer
93 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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2answers
996 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
41 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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1answer
2k 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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127 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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44 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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184 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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44 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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1answer
121 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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179 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
47 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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25 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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53 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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9answers
2k views

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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217 views

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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700 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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135 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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73 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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43 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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26 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?
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42 views

Can this equation have more than one solution?

Consider the following equation: $\left[\array{1 & 0.1353 & 1 \\0.3678 & 0.3678 & 1 \\ 0.1353 & 1 & 1 \\ 0.3678 & 0.3678 & ...
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57 views

Solve inverse tangents

How do I solve the following equation: $$ \tan^{-1}\frac{x}{10^6}+\tan^{-1}\frac{x}{10^7}=90^{\circ}$$ WA Step by step solution from wolframalpha is unavailable.
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31 views

Generating other left inverses of a matrix

I have a non-square matrix $G$ and I am looking for matrices like $F$ such that $FG=I$. I am told that it has not a unique solution. I calculated a (left?) inverse of $G$ using the formula ...
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58 views

Inverse Laplace transform $\mathscr{L}^{-1}\{ \frac{1-1e^{-2s}}{s(s+2)} \}$

I am trying to calculate the following inverse Laplace transform. I tried to apply partial fraction decomposition to make it easier to take the inverse but it doesn't seem to work, $s$ is a power in ...
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Subring of the field of rational numbers

Let $R=\{a\cdot2^n\mid a,n \in \mathbb{Z}\}$ be a subring or the field of rational numbers $\mathbb Q$. i) What kind of elements are invertible in $R$? ii) Prove that $R$ is a principal ...
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47 views

The differentiability class of the inverse function

Here's the final part of a proof (from Marden's Elementary Classical Analysis) of the inverse function theorem, where we have been given that $f$ is of class $C^p$: Could someone please explain the ...
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156 views

Proof for diagonalizable matrix

Let $A \in M_n(\mathbb C)$ be invertible. Prove that $A$ is diagonalizable if and only if $A^{-1}$ is diagonalizable. This is what I have for one direction of the proof: Suppose $A$ is ...
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1answer
140 views

Finding the multiplicative inverse of an element in $\mathbb Q[x]/(x^3-2)$

I have a problem here that asks: "Express the multiplicative inverse of $1+2^{1/3}-3\cdot2^{2/3}$ as $a_0+a_1\cdot2^{1/3}+a_2\cdot2^{2/3}$." I believe they are asking us to find it by utilizing the ...
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107 views

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

I want to show that f(x) is bijective and calculate it's inverse. Let f (x) : R → R be defined by f (x) = (3x/5) + 7 I understand that a bijection must be injective and surjective but ...
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81 views

Inverse CDF of a Standard Normal Variable

In many applications, for example Monte Carlo methods, we require the inverse CDF of a standard normal random variable. But the CDF: $$ \Phi(x)= \int_{-\infty}^ x \frac{1}{\sqrt{2\pi}} e^{-t^2/2} dt ...
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Continuous function that is invertible in one argument---is its inverse continuous in both arguments?

Suppose that $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous function and that it is invertible in its second argument, i.e. for every $x \in \mathbb{R}$, $f(x,\cdot)$ is invertible with ...
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3answers
201 views

Lower triangular matrices [duplicate]

Is the inverse of an invertible and lower triangular matrix still both lower triangular and invertible?
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1answer
90 views

An invertible matrix

Given Matrix $A$, checking that its diagonal elements are nonzero or whether its determinant is nonzero, can we say the matrix is invertible for sure? Are there other properties that by looking at the ...
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1answer
116 views

Iterative update of pseudo inverse solution

I have an overdetermined linear problem of the form $A x = b$, which is solved in least squares sense using the Moore–Penrose pseudo invers. The issue now is, that over time additional constraints and ...
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26 views

Finding the correct angle from inverse cosine?

For my math homework, I have to find an angle of rotation, $\theta$, by cos $\theta$ = $-\sqrt3/2$. When I plug this into my calculator, I get 5$\pi$/6, but the correct answer is -5$\pi$/6. What is ...