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 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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40 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
169 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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63 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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742 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
38 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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113 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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42 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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1answer
160 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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108 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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156 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
43 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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0answers
24 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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50 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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214 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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1answer
500 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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123 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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71 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
39 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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41 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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56 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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30 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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53 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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90 views

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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1answer
43 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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127 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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131 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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73 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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69 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
163 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
83 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
90 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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1answer
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 ...
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29 views

Inverse of a matrix expression after SVD

Let: $UDV^T$ be the SVD decomposition of $A$; $\lambda\in\mathbb{R}$ and $I_n$ the identity matrix Why is the following true? $(VD^2V^T+\lambda I_n)^{-1} = V(D^2+\lambda I_n)^{-1}V^T$
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1answer
83 views

Least Square with homogeneous solution!

I've read somewhere that: $x=A^+b+(I-A^+A)Z$ is a solution for $Ax=b$ ,when is doesn't have a particular solution. where $A^+$ indicates the pseudo-inverse and $Z$ is an arbitrary vector!!! I know ...
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2answers
36 views

Show that this is one to one continuous and find its inverse which is continuous as well.

Let's define $\phi: \Bbb R^2 \to S$ for $S$ is subset of $\Bbb R^3$ For constant $a,b,c,d$ and $c\not =0$ $$\phi(x,y)=(x,y, \frac{d-ax-by}{c})$$ I want to show that the function $\phi$ is 1-1 ...
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1answer
45 views

Stuck at Extended Euclidean Algorithm to solve equation

I'm trying to solve the following function via the Extended Euclidean Algorithm, but I'm stuck at the last step where I need to sub in sub 2. ...
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57 views

Example of sets $A$ and $B$ and functions $F$ and $G$ such that $F: A \rightarrow B, G: B \rightarrow A, G \circ F = I_{A}$, and $G \neq F^{-1}$

Give an example of sets $A$ and $B$ and functions $F$ and $G$ such that $F: A \rightarrow B, G: B \rightarrow A, G \circ F = I_{A}$, and $G \neq F^{-1}$ I was thinking maybe $F$ can be a function ...
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Inverse of a 4x4 matrix with variables

I missed my class on the inverses of matrices. I'm catching up well, but there's a problem in the book that got me stumped. It's a 4x4 matrix that is almost an identity matrix, but the bottom row ...
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1answer
27 views

Simplify functions involving modular arithmetic

In this question, the answer says that f o g(x) = x. But I am unable to get this result. The expression I am able to get is that f o g(x) = 7*(x mod 3) + 57*(x mod 7) (mod 21). I am unable to ...
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47 views

Inverse Laplace transform of ratio of polynomials

I would like to understand if the inverse laplace transform of these functions gives something in terms of known functions (such as exponentials..see my examples). ...
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1answer
31 views

Inverse of function arcsin

I'm having trouble finding the solution of the inverse of the function ${\rm f}\left(y\right) = \arcsin\left(\,3 - x2\,\right)$ Isn't $\arcsin$ the inverse of $\sin$ ?. This is what I have now as ...
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1answer
26 views

Parametric Curves Existence of Tangent

If $\frac{dy}{dt}$ and $\frac{dx}{dt}$ exist, then does $\frac{dy}{dx}$ always exist when $\frac{dx}{dt} \not=0$? Indeed, this is a very simple question. Sorry but I'm just a beginner for ...
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49 views

How to calculate the inverse of the line integeral.

Let $f$ be a polynomial function, $$ f(x) = a_0 + a_1 x + ... + a_d x^d $$ where $a_0$, $a_1$, ..., $a_d$ are parameters and usually $d \le 6$. Let $g$ be the line integral of $f$, $$ g(x) = ...
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3answers
126 views

Finding inverse of a matrix

This question is in my assignment. We are not allowed to use any symbol to represent any elementary row and column operations used in the solution. We must solve it step-by-step. Please help me to ...