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 ...
2
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4answers
153 views
is this function invertible ??
given the function
$$ f(x)= x+\cos(x)+\sin(\cos(x)) $$ (1)
is this invertible ?? i mean it exists another function $ g(x) $ so
$$ f(g(x))=x $$
my guess is that for big $ x \gg 1 $ the function ...
1
vote
2answers
86 views
What does this syntax mean: “$f^{-1} : N_{10} \Rightarrow N_b $ is the inverse of $f: N N_{b} \Rightarrow N_{10}$?”
I'm trying to solve this but I haven't seen syntax like this before. Can someone please explain the syntax?
http://i.imgur.com/GO1Ki.png
The image is
Show that the one-to-one function $f^{-1} : ...
4
votes
1answer
142 views
Inverse function notation
Suppose $f$ and $g$ are functions that fail to be one-to-one, but $f+g$ is one-to-one. Has anyone ever seen the notation $(f+g)^{-1}$ for the inverse function in that situation? (I find myself ...
1
vote
1answer
64 views
A differentiable injective function with Lipschitzian Inverse
I'm having difficulty with the following question which was given to me following studying the inverse mapping theorem.
Let $U\subseteq\mathbb{R}^{n}$
be an open set and let $f:U\to\mathbb{R}^{n}$
...
0
votes
1answer
80 views
Help with restricted domain of a function to find inverse
Restrict the domain of $f(x)$ to find inverse:
\begin{align}
f(x) & = x^2+6x+9 = (x+3)^2 \\
g(x) & = \sqrt{x} - 3
\end{align}
0
votes
2answers
519 views
To invert a Matrix, Condition number should be less than what?
I see that there is a matlab tag in this site, so I ask my question here and not in stackoverflow although it is also related to programming in matlab.
I am going to invert a positive definite matrix ...
4
votes
2answers
108 views
$\ln(x)$, $e^{x}$ and $\int \frac{1}{x}dx$ relationship
My math professor told me that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$ by the definition; so far so good.
But how/why does $\ln(x)$ ($\int_1^x\frac{1}{t} dt$: by defintion) coincide with the inverse of ...
0
votes
2answers
122 views
$e^x-x-4$equating with zero
I want to find out the values of x where the $f(x) = e^x-x-4$ will equal zero.
My problem by solving this myself is that I cannot use logarithm natural (ln) because I have a normal x:
$f(x) = e^x - ...
2
votes
1answer
74 views
For square matrices $A$, $B$, is $AB=I$ sufficient that $A$ and $B$ are inverse of each other? [duplicate]
Possible Duplicate:
If $AB = I$ then $BA = I$
If $A$ and $B$ are two square matrices, and we know $AB=I$ where $I$ is the identity matrix. Is it sufficient that $BA=I$ as well so that $A$ ...
1
vote
3answers
89 views
Is this an invertible linear transformation?
Suppose you have a linear transformation $T: M_{2\times 2}\to M_{2\times 2}$ given by
$$ \begin{pmatrix} a & b \\ c & d\end{pmatrix}\mapsto \begin{pmatrix} a+b & a \\ c & ...
0
votes
1answer
34 views
Error bound for pseudoinverse
Hi I have a generic matrix A, is it possible to bound the error defined as $\|A^+A−I\|$ ??
Are there some reasonable assumptions (es. random matrix, etc...) I can make in order to have a better bound ...
0
votes
1answer
16 views
Should I check Multicollinearity When There is An Inverse?
At Machine Learning algorithms there are usually inversion process about matrices and sometimes Matlab throws error when Multicollinearity occurs.
Should I check Multicollinearity(and how) everytime ...
2
votes
1answer
111 views
Finding inverse of a function that is mixture of exponentials
How can we find the inverse of this function?
$$y=\exp(ax)+\exp(bx),$$
where $a$ and $b$ are constants
0
votes
1answer
171 views
Method of finding inverse of a Matrix using minimal polynomials
Using a piece from my last question I want to show how to find $A^{-1}$ as a polynomial expression in $A$ of degree < $\deg m_A$ where the leading coefficient of the polynomial is ...
3
votes
4answers
109 views
How do you take the multiplicative inverse of a p-adic number?
I am reading the wiki page for p-adic numbers and it states that they are a field extension of the rationals so each member has to have a modular multiplicative inverse.
So how would I take the ...
1
vote
2answers
112 views
second derivative of the inverse function
I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$
But how to derive the formula for the second derivative of g(y) knowing that $[\frac{1}{f(x)}]' = ...
2
votes
1answer
149 views
Computing inverse two-sided Laplace transform symbolically
How can I compute the inverse two-sided Laplace transform symbolically?
I know MATLAB has ilaplace[1], but that's just for a one-sided transform.
[1] ...
1
vote
2answers
404 views
Find inverse of exponential function
Do you know how I could compute the inverse function of the following exponential sentence?
$$y=\dfrac{e^x}{1+2e^x}$$
1
vote
1answer
774 views
Find inverse of polynomial function
Do you know how I could compute the inverse function of the following polynomial?
$f(x) = x^5+x^3+x$
Thanks in advance.
2
votes
2answers
58 views
Modular Arithmetic Equations
I'm trying to solve $x^{16} = [1]_{989}$ in $x ∈\Bbb Z/989\Bbb Z$.
I tried a few simplifications but don't know how to solve it.
Any help is welcome.
2
votes
0answers
237 views
Multivariable Inverse Function problem
Consider the system of equations
$$\left\{\begin{align*}
&x^5 v^2 + 2y^3 u = 3\\
&3yu - xu v^3 = 2\;.
\end{align*}\right.$$
Show that near the point $(x,y,u,v) = (1,1,1,1)$, this system ...
0
votes
0answers
59 views
on norm of submatrix of the inverse and inverse of a submatrix
Given an M-matrix, say $M\in\mathbb{R}^{n\times n}$, which in block form is
$M$ = \begin{pmatrix}
A & B\\
C & D
\end{pmatrix}
where $A\in\mathbb{R}^{k\times k}$ and ...
1
vote
3answers
129 views
Why isn't every coproduct a product (and vice-versa)?
So I know that every coproduct is not a product, so I am misunderstanding some part of the definition of (co)products. Saying that U is a coproduct (the disjoint union of X1 and X2 below) of objects ...
4
votes
0answers
116 views
Inversion of elliptic integral
I have an equation of the type
$$
p=\int_0^b\sqrt{\left(a^2-x^2\right)\left(b^2-x^2\right)}dx,
$$
in which $a$ and $b$ (with $a>b>0$) are (known) functions of some parameter $H$ (such that it is ...
1
vote
3answers
594 views
Proving that the matrix is not invertible.
A is a 2x3 matrix and B is a 3x2. How can i prove that the matrix D = AB is not invertible. I could not go further in this problem. The only thing that i have found is the multiply of these two matrix ...
1
vote
1answer
78 views
Inverse of trace class operator restricted to it's range
A paper I'm reading constructs the Cameron-Martin space in a way different than I'm used to, and in the process they gloss over a functional analysis result about the existence of an inverse. It ...
3
votes
2answers
91 views
Find inverse function
Is it possible to get inverse of all be functions? For example, can we calculate inverse of $y=x^3+x$?
1
vote
1answer
56 views
Invertibility of matrix with each element equal to cofactor
I am doing an exercise book which has one problem that asks you to prove the nonsingularity of a matrix if each element of the matrix equals its cofactor (the determinant submatrix by deleting the ...
1
vote
3answers
70 views
What is the relation between two invertible functions
Lets say that if f(x) and g(x) are invertible.
1- is (f(x)+g(x)) also invertible?
2- is f(g(x)) invertible too?
for the first one lets say that
f(x)=x and g(x)=-x
then f(x)+g(x)=x+(-x)=0
and ...
1
vote
3answers
72 views
Determinant of a $4\times4$ invertible matrix
Let $A$ be a $4$ by $4$ invertible matrix, such that $\det(3A)=3\det(A^4)$.
Then $\det(A)=3$.
Would somebody please give me some clues on this?
Thanks
5
votes
0answers
214 views
Functions whose derivative is the inverse of that function
Everyone knows that there are at least three functions whose derivative is the function itself, namely $e^x, \ 0$ and $-e^{x}$. ( are there more?)
I was drawing some polynomials and their ...
2
votes
2answers
163 views
Trigonometric general solution to ordinary differential equation
Solve: $$\frac{dx}{dy}=(x^{2}-x-12)(1+\tan^{2}{y})$$
This is a first order, linear, separable ODE, so it can be solved by rearranging to:
$$\frac{dx}{x^{2}-x-12}=(1+\tan^{2}{y})\:dy$$
And then ...
0
votes
0answers
18 views
$\dfrac{f_2(-g_2(z))} {g_2(-f_2(z))} = h(z) + O(1/z)$
Let $z$ be a complex number.
Let $h(z)$ be a given function.
We are looking for functions $f(z),f_2(z),g(z),g_2(z)$ such that :
$f(z)$ and $g(z)$ are eachothers functional inverse.
$f_2(z)$ and ...
1
vote
2answers
76 views
inverse of a binomial matrix
I have a matrix $A$ ($n \times n$) defined as follows:
$$A = \{ 0 \text{ if } i<j,\ \mathrm{Binom}(x=i, \mathrm{size}=j, \mathrm{prob})\text{ if } j \ge i\}$$
This is an upper triangular matrix, ...
0
votes
1answer
60 views
Inverse of a Function of Random Variables
I'm hoping to get a hint on a problem.
The problem formulation is: there are two random variables X and Y, both of which are a Uniform RV on (0,1).
Let x be values on (0,1) for X and y be values on ...
0
votes
2answers
99 views
how to find two right-inverse functions of a function
i am stuck in this problem. i need to find two right-inverse functions of this function:
$h: \Bbb N_0\times \Bbb N \to \Bbb N, (m,n)\mapsto m+n$.
i know that the function h' is a right inverse of ...
0
votes
2answers
53 views
Find the inverse of $\alpha^{38}$ in $\mathbb F = \mathbb Z_2[x]/\left<x^4+x+1\right>$
Let $\alpha$ be a root of $x^4+x+1$ and we are given some powers of $\alpha$ as linear combinations of $1,\alpha,\alpha^2$ and $\alpha^3$
$\alpha^4=\alpha+1$
$\alpha^5=\alpha^2+\alpha$
... (the rest ...
0
votes
3answers
54 views
How to find the inverse of integer $i$ in $\mathbb Z_{n}$
In my understanding, a number $i$ has an inverse $i^{-1}$ in $\mathbb Z_{n}$ if $i\times i^{-1} \equiv 1 \pmod{n}$
e.g.: In $\mathbb Z_{14}$ the inverse of $3$ is $5$ since $3\times5\equiv1\pmod{14}$
...
3
votes
1answer
79 views
Pseudo inverse not equal inverse — conditions?
What are the conditions under which a the pseudo-inverse of a matrix is not equal to its inverse?
I have a matrix equation:
$$
AXB = C
$$
which according to Laub (13.14, 13.15) has a solution if
...
4
votes
0answers
42 views
Is the inverse of any elementary function asymptotic to some elementary function?
Is the functional inverse of any elementary function asymptotic to some elementary function ?
For instance Lambert's $W(z)$ is asymptotic to $ln(z)$. See ...
1
vote
3answers
252 views
Finding The Equivalence Class
Okay, so the question I am working on is, "Suppose that A is a nonempty set, and $f$ is a function that has A as its domain. Let R be the relation on A consisting of all ordered pairs $(x, y)$ such ...
2
votes
0answers
290 views
inverse of a covariance matrix 3x3
I have 2 pixels with size 1x3 called $A$ and $B$ and I have to compute the following equation:
$$
A^T *(\Sigma+ I_3*\lambda)^{-1}*B
$$
where $\Sigma$ is the covariance matrix (3x3) between vectors ...
2
votes
1answer
153 views
Linear Algebra Question ( rank of matrix )
Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively
prove $\operatorname{rank}(\mathbf{PA}) = ...
0
votes
1answer
52 views
Inverse of a function
From my text book it says that $f(x)= x^3$ and $f^{-1}(x) = \sqrt[3]{x}$ , which I totally agree with.
why does $f(x)= 1/(x-1)$ and $f^{-1}(x)= 1/x + 1$ and not equal $f^{-1}(x)= 1/(x+1)$?
I know ...
1
vote
1answer
72 views
How to find the frechet derivative at $A\rightarrow A^{-1}$ mapping?
I am reading on my own the Lectures on the Geometry of Manifolds (http://nd.edu/~lnicolae/Lectures.pdf ) , and got stuck in solving the exercise 1.1.3 (b) .
The 1.1.3 (b) is :
Let F: $U\rightarrow ...
1
vote
2answers
174 views
1
vote
2answers
58 views
Calculate the inverse of $ s(x) = \frac{1+f(x)}{1-f(x)}$ in terms of $f^{-1}$
Calculate the inverse of $s(x) = \frac{1+f(x)}{1-f(x)}$ in terms of $f^{-1}$,f is a $1-1$ function with inverse $f^{-1}$
0
votes
1answer
24 views
intuitive explanation of sparsity / references
I know it is a vague question, but I am confused by why/when we actually want sparsity of a matrix. For example, interior-point methods work better when constraint matrix is sparse. Similarly, it is ...
0
votes
0answers
153 views
How do we find the inverse of a function $f(m,n)$ if there is a constant k?
I know I need to use the inverse matrice, but the problem is (the parameter) $k$, because it's a variable that can take any value depending on $k$, but it's not a variable. Think of the derivative ...
2
votes
3answers
473 views
How do we find the inverse of a function with $2$ variables?
$$f(m,n) = (2m+n, m+2n)$$
What do we have to do to find the inverse of this function?
I don't even know where to begin.
