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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Derivative of trace of inverse of a matrix function

I am trying to derive the derivative of the trace of inverse of a matrix function (of X), i.e. $$f(X)=Tr\left((HXH^{H}+I)^{-1}\right) $$ where $H\in R^{n\times m}, X\in R^{m\times m}$. So ...
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18 views

Polynomial function for arctan(tanx) [on hold]

What is the Equivalent polynomial function for arctan(tanx), arccos(cosx), arcsin(sinx)?
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3answers
842 views

from weighted average to single values

think about when you're computing a weighted average: you do something like W = sum(amount*weight) / sum(weight) Now I need to find every amount (amount0, amount1.....) starting from knowing the ...
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1answer
25 views

What is the inverse of the function $f: x \mapsto (x,x^{2}) : \mathbb{R} \to \mathbb{R}^{2}$?

Let $f: x \mapsto (x,x^{2}) : \mathbb{R} \to \mathbb{R}^{2}$ and let $Y := f(\mathbb{R})$. Then $\mathbb{R}$ and $Y$ are in injection via $f$. Moreover, since $Y$ is the range of $f$, certainly ...
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2answers
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Transpose of inverse vs inverse of transpose

I can't seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix? Thanks!
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35 views

$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse?

Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) ...
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1answer
19 views

Matrix and eigenvalues question hints?

This is the homework I have done part a, b, but I don t have any idea how to do the rest $y = 5$ and $z = 12 $ Those are the eigenvalues of matrix $A$ For part c, and d, I've tried to put some ...
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8 views

Inverse of the sum of a symmetric positive definite matrix and a diagonal (but with different entries) matrix

Suppose we have symmetric positive definite $A$ with the size of $d \times d$, giving the SVD $A=V\Sigma U^T$ , if $D$ is an identity matrix, ie $D=I$, then $(A^T A + \gamma I)^{-1}=U (\Sigma^2 + ...
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12 views

Solutions for the dependency problem

Currently I read about the dependency problem of interval arithmetic. Mainly it's the problem that in the equation $X-X$ for $X$ being an interval the following is calculated: $$X-X=\{x-y:x\in X, y\in ...
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3answers
59 views

Geometric interpretation of inverse complex function?

Function $f\colon\mathbb{R}\to\mathbb{R}$ and its inverse $f^{-1}$ are symmetric over line $y=x$. It's easy to imagine inverse of real function, we just have to "flip" the plot over $y=x$. But what ...
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26 views

Solve the system of trigonometric equetions, inverse kinematics

I am trying to do inverse kinematics for some mechanical system. After applying Neton-Euler method following equations were obtained: $$F_x = k_f w_l\sin(\beta_l) + k_f w_r\sin(\beta_r)$$ $$F_y = k_f ...
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79 views

If $f^{-1}(x)$ is continuous, is $f(x)$ also continuous?

Let $f:\mathbb{R}\mapsto\mathbb{R}$ be a one-to-one function with $f(\mathbb{R})=\mathbb{R}$. If $f^{-1}(x)$ is continuous $\forall x\in\mathbb{R}$, prove or disprove that $f(x)$ is continuous ...
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1answer
34 views

Inverse sum representation of sine

The other day I was playing with functions of the form $$ f(x) = \frac{1}{\frac{1}{a_0(x-b_0)} + \frac{1}{a_1(x-b_1)} + \cdots + \frac{1}{a_n(x-b_n)}} $$ and I found particularly that $$ ...
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1answer
23 views

Cayley transform a matrix that is invertible when added to the identity

Let A be an nxn matrix such that (I+A) is invertible. I need to prove that the Cayley Transform of A, denoted by $A^c$, is such that $(I+A^c)$ is invertible. The Cayley Transform is defined as ...
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2answers
30 views

Geometric Interpretation of Determinant of an Inverse Matrix

The $\mathbf{A}$ be an $n\times n$ full rank matrix. Then, the (signed) volume enclosed by the rows (or columns) of $\mathbf{A}$ is equal to $\det(\mathbf{A})$. My question is, what is a geometric ...
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127 views

Proof of the Inverse Function Theorem using the Contraction Mapping Principle.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
3
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1answer
45 views

Reciprocal of the reciprocal of zero

By straightforward evaluation, $$(0^{-1})^{-1}=(NaN)^{-1}=NaN$$ where $\frac{1}{0}$ is taken to equal $NaN$ (not a number), or undefined or indeterminate. However, the laws of exponents state that ...
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1answer
19 views

Is the condition “the inverse image of a closed base set is closed?” sufficient for continuity?

Let's say you have a function $f:X \to Y$, where $X$ and $Y$ have topologies. The set $C$ forms a closed base for $Y$. If for every $c \in C$, $f^{-1}(c)$ is closed in $X$, is $f$ continuous? If the ...
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15 views

A class of sparse matrices whose inverse is also sparse?

In general the inverse of a sparse matrix is dense. A notable (but trivial) exception from that rule are diagonal matrices. Is there any other (broad) class of sparse matrices whose inverse is also ...
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Why does this “miracle method” for matrix inversion work?

Recently, I answered this question about matrix invertibility using a solution technique I called a "miracle method." The question and answer are reproduced below: Problem: Let $A$ be a matrix ...
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1answer
446 views

relation between size of matrix and condition number

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
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22 views

Finding multiplicative inverse Euler's theroem

been struggling this whole day with trying to figure out the multiplicative inverse of 17 modulo 31 using Eulers theorem. We know that 31 is a prime, φ(n)=30, so i end up with 17^30=(cong)1 (mod 31). ...
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45 views

Is the linear operator $T(f)(x) = f(-x) + f(x)$ invertible?

My understanding is that the inverse of a linear operator will effectively "undo" the operation. Therefore to get the inverse of this operator we need to somehow subtract the f(-x). But I'm not ...
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1answer
31 views

Is the linear operator $T_f(x) = f'(x)$ invertible?

I think that $T_f(x)=f'(x)$ is invertible. This seems likely because it is a differential operator, and the inverse of a differential operator is the integral operator (though I'd like more ...
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2answers
301 views

Prove (local) converse to the implicit function theorem

The implicit function theorem tells us that: Given a level set $M^k = F^{-1}(F(p_0))$ of a smooth function $$F: \mathbb{R}^n \to \mathbb{R}^{n-k},$$ where $\operatorname{rk}{(Df)(p)} = n-k$ for ...
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1answer
20 views

Condition inverse $p$-adic number

Take $p$ prime, $n \in \mathbb{Z}_{>0}$ and $x \in \mathbb{Z}_p$. Suppose that $p$ isn't a divisor of $$x = (x_j + p^j \mathbb{Z})_{j \in \mathbb{Z}_{>0}},$$ then one can prove that the first ...
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63 views

Find inverse of 15 modulo 88.

Here the question: Find an inverse $a$ for $15$ modulo $88$ so that $0 \le a \le 87$; that is, find an integer $a \in \{0, 1, ..., 87\}$ so that $15a \equiv1$ (mod 88). Here is my attempt to answer: ...
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326 views

How to find inverse of 2 modulo 7 by inspection?

This is from Discrete Mathematics and its Applications By inspection, find an inverse of 2 modulo 7 To do this, I first used Euclid's algorithm to make sure that the greatest common divisor ...
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30 views

$A,B$ are matrices $3x3$ so that $B^2A=-2B^3$ and $B^3+AB^2=3I$ express $A^{-1}$ and $B^{-1}$ using $B$

I have the follow question : $A,B$ are matrices $3x3$ so that $B^2A=-2B^3$ and $B^3+AB^2=3I$ express $A^{-1}$ and $B^{-1}$ using $B$ I tried to "play" with the equations but I always get stuck with ...
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1answer
13 views

Understanding Multiplicative Inverse in RSA

Okay so I am reading up on RSA, trying to understand how it works, and I come across this $ x∈ℤp, x−1 ∈ℤp ⟺ \gcd(x,p) = 1$ Now it then gives an example, as follows: Lets work in the set $ℤ9$, ...
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17 views

Finding the $h'(x,y,z)$ if $h= p \circ q $ $p(x,y,z)=(x \sin y, x \cos y, z+y ), q(x,y,z)=(x^2,x+y,2e^z)$

I just want someone to check my work basically. Providing thoughts and insight, into possible mistakes: Finding the $$h'(x,y,z)$$ if $$h= p \circ q ,\ \ p(x,y,z)=(x \sin y, x \cos y, z+y ), \ \ ...
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2answers
489 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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3answers
260 views

Find all matrices where the matrix is its own inverse and the determinant is 1

I need to find all the matrices: $$\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ such that $$ad-bc=1$$ and $$A^{-1}=A$$ How would I go about doing this? I know that $AA=I^2$, ...
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0answers
12 views

Inverse of a product of real functions

Given $F(x) = L(x)G(x)$, with $L$ and $G$ real function strictly greater than zero. Suppose that F and G are decreasing functions (so that $F^{-1}$ and $G^{-1}$ exists). What can we say about the ...
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47 views

Why can't this mixed function be inverted?

Given the function $$y=Ax + B\sqrt x$$ where $A$ and $B$ are real constants, $x$ is real and $x > 0$ I want to find the inverse where $x$ is a function of $y$. ButI don't believe that's possible ...
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2answers
61 views

Inverting a $3\times 3$ block matrix

Suppose that $a$ and $b$ below are scalars, $F$ a square matrix, $v$ a column vector. I'm trying to invert the matrix $M$ of the form $$ M=\begin{pmatrix} a & v' & 0\\ v & F & 0\\ 0 ...
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0answers
40 views

What is Lebesgue measure of sets of inverse prime numbers in $[0,1] $?

I would like to know if it is possible to know the lebesgue mesure of sets of inverse prime numbers in $[0,1]$ Note : I think should to know in the first if the sets of primes are infinit countable ...
2
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1answer
24 views

Block matrix pseudoinverse: symmetry of the inverse of a symmetric matrix

In the wiki page for block matrix pseudoinverses, there is a formula $$ \begin{pmatrix}A & B \\ C & D\end{pmatrix}^{-1}=\begin{pmatrix} (A-BD^{-1}C)^{-1} & ...
2
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1answer
113 views

Is there an efficient method to find all the self-inverse matrices with integers in a given range?

Given $n$ and a range, for example $[-10,10]$, is there an efficient method to find all $n \times n$-matrices $A$ with integers in the given range, which are self-inverse (that means the equation ...
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26 views

Flip Values to get the opposite

Not sure of the name of what I need to do, but I used to do it all the time, and now i forget. I have values 1 - 10. I want 10 to become 1 and 1 to become 10. What is the formula to do this again? It ...
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1answer
48 views

L2 norm of an inverse of a sum of matrices

I am trying to take the L2 norm of the following expression: $-(H^{-1} + bI)^{-1}v$, where $H$ is a psd matrix, b is a scalar, and $v$ is a vector. In particular I am having trouble with the first ...
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1answer
414 views

Cholesky decomposition of the inverse of a matrix

I have the cholesky decomposition of a matrix M. However, I need the cholesky decomposition of the inverse of the matrix, invM. Is there a fast way of getting this, without first inverting the matrix ...
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0answers
16 views

Finding the inverse of a square circulant matrix

I'm having difficulties of finding the inverse of the following square matrix, which consists of $2\times 2$ circulant matrices: $A = \left[\begin{matrix}x^{383} & x^{102} + x^{253} \\ ...
3
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1answer
55 views

Inverse of a matrix and its transpose

I'm trying to figure out why the calculation below works. I do know that $(A^T)^{-1} = (A^{-1})^T$. The matrix A = $\begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1 ...
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1answer
22 views

Calculate the modular inverse of $2a$ given that of $a$

My problem is that I have to calculate some modular inverses of numbers that are related by multiplying by $2$, that is: Given $a$ and $x$ so that $ax\equiv1\mod n$ ($n$ being an odd number) I need ...
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77 views

Can we prove $BA=E$ from $AB=E$? [duplicate]

I was wondering if $AB=E$ ($E$ is identity) is enough to claim $A^{-1} = B$ or if we also need $BA=E$. All my textbooks define the inverse $B$ of $A$ such that $AB=BA=E$. But I can't see why $AB=E$ ...
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1answer
547 views

Efficient Cholesky decomposition of inverse matrix

I want to generate random numbers from a multivariate normal distribution in Matlab. Normally, this is done like: $w = \overline{w} + \text{chol}(\Sigma) \cdot \vec{l}$ But in my case I don't know ...
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3answers
40 views

If a function has an inverse then it is bijective?

I have some trouble finding the answer to this, can someone help me out: If I have a general function $f$ with domain $X$ and codomain $Y$, I know nothing about the function (injective, surjective). ...
2
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1answer
546 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
62 views

Is there a polynomial $p$ such that it is bijective and $ p: \mathbb{Q}^n \rightarrow \mathbb{Q}$ for $ n>1$ ??

Let us define a polynomial $p$ defined as follow $$p: \mathbb{Q}^n \rightarrow \mathbb{Q}.$$ I acrossed this question in mind. Is there a polynomial $p$ such that it is bijective and $p: ...