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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56 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
126 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
108 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
519 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
69 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
90 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
52 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
66 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
71 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
195 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
151 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 ...
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votes
0answers
16 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
72 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
57 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
96 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
51 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
52 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
76 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
39 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
214 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
283 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
138 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
47 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
69 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
158 views
1
vote
2answers
56 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
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0answers
137 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
388 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.
1
vote
3answers
81 views
Is the inverse of a function the reflection of the function about the line $y=x$?
So if we have
$f(x)$:
$y=x$ when $x \ne 1$ and
$y = 0$ when $x = 1$.
The inverse would be:
$y=x$ when $x \ne 0$ and
$y=1$ when $x = 0$
?
0
votes
4answers
69 views
Invertibility of the operands of an invertible product of matrices
Let $S$ and $T$ be two matrices. Assume that $ST$ is invertible. I want to prove that $S$ and $T$ are invertible.
I managed to prove that $T$ is invertible. Here is my proof. Assume $T(X)=0$. Then ...
3
votes
3answers
192 views
How to invert this function? (Inverse exponential function with arctan)
How to invert this function?
$$
y = e^{\arctan(x^5)}
$$
1
vote
1answer
104 views
Cross product and inverse of a matrix
I would like to show that $\left(\begin{array}{ccc}
1 & s & s^2 \\
1 & t & t^2 \\
1 & u & u^2
\end{array}\right)$ has an inverse provided $s$, $t$ and $u$ are distinct.
I ...
1
vote
0answers
45 views
Solving inverse square of visible scale
I'm not super-adept in mathematics, so I turn to you for help. As I read, the perceived scale of an object reduces by the inverse square as the viewed distance increases. In order to solve for this, ...
1
vote
2answers
96 views
How to prove my statement is correct about invertible matrices?
SO far i showed that if A matrix is left invertible (L) then in Ax = b, x has at most 1 solution.
I got that LAx = x = Lb, so x = Lb
for right inverse (R) of A, in Ax = b, x has at least one ...
0
votes
2answers
392 views
How to find the inverse to $f(x)= x^2 - 6x + 11 $
If the inverse exists, how do I find the inverse to this function:
$$
f(x)= x^2 - 6x + 11
$$
with $x \le 3$
Stuck at the quadtric formula. I think i have got the right answer which is $x = 3 ± ...
1
vote
1answer
98 views
inversion of the function $ \sqrt x \ln x $
is there an EXACT (not asymptotic) inversion of the function $ \sqrt x \ln x $ or can we only obtain this inverse in terms of a power series ?? thanks.
4
votes
2answers
168 views
Is $a^{-1} + b^{-1} = (a + b)^{-1}$ always true for Abelian group?
I get the equation $a^{-1} + b^{-1} = (a + b)^{-1}$ from ordinary + operation. For ordinary + operation I mean $a^{-1} = -a$. It is also true for * of rational numbers $3^{-1}*4^{-1} = \frac{1}{3} * ...
2
votes
2answers
437 views
Inverse function of a polynomial
What is the inverse function of $f(x) = x^5 + 2x^3 + x - 1?$ I have no idea how to find the inverse of a polynomial, so I would greatly appreciate it if someone could show me the steps to solving this ...
0
votes
1answer
171 views
Eigenvalues of a Toeplitz matrix
A toeplitz matrix $X$ is given as,
\begin{equation}
X =
\begin{pmatrix}
~~~~\textbf{1} ~~~~\textbf{c} ~~~~\textbf{d} ~~~~0 ~~~~0 ~~~~0 ~~~~\textbf{d}~~~~ \\
~~~~\textbf{c} ~~~~\textbf{1} ...
1
vote
1answer
69 views
Inverse image of disjoint is disjoint?
If I have two sets that are disjoint i.e. $A\cap B=\emptyset$, and $\varphi \in C^1(U,\mathbb{R}^N)$, then are the inverse images (i.e. $\varphi^{-1}(A), \varphi^{-1}(B)$) also disjoint?
My logic ...
1
vote
1answer
115 views
Terminology question; inverse vs complement in Boolean algebra
This was said at a lecture I attended:
$e$ is neutral element for operation $*$ if $\forall x (x*e=x \wedge e*x = x)$.
So, for example 0 is n. e. for disjunction and 1 is n. e. for ...
3
votes
1answer
108 views
Non-negative matrix and inverse
Lately, I´ve been struggling with math homework and came across a question I´m not sure how to answer. I will be glad for any help...
Suppose we have matrix $A$ (size $n\times n$) and its inverse ...
0
votes
1answer
96 views
Equivalence Relations with inverses?
I have no Ida how to approach this problem:
Suppose S is a relation on a set X which is reflexive and transitive. Then S intersection S inverse is an equivalence relation on X.
Any idea on how I ...
2
votes
1answer
71 views
Invertible Derivative
I'm trying to brush up on some differential geometry, but there's a subtle point I don't understand. Suppose $h$ is a diffeomorphism. Then the lecture notes here suggest that it's derivative $df_x$ is ...
0
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1answer
181 views
inverse,multiplicative inverse and Congruence of a prime field
I am dealing with ECC in these days which heavily based on finite fields. I want to how to find a inverse of a value in finite field and what is multiplicative inverse and also Congruence
F29-
...
1
vote
0answers
39 views
Looking for correct terminology
Given the functions $f\colon A\to B$ and $g\colon B\to B$, a common, useful strategy is to define a new function $h\colon A\to A$ as the composition $f^{-1}\circ g\circ f$.
There seem to be many ...
0
votes
1answer
41 views
How is it simplified?
1 ) I have this equation and don't know how $(1 + 2e^y)$ and $(1 + 2e^x)$ from each side of equation cancelled each other out and get the final answer $x = y$.
...
1
vote
1answer
56 views
Conditions on function inverses
I have recently asked a question related to an inverse function which was not so obvious to calculate:
Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$
Now I would like to learn;
Given ...
5
votes
3answers
268 views
Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$
I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given:
$$y=W(e^{ax+b})-W(e^{cx+d})+zx$$
where $W$ is the Lambert $W$ function and ...
