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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votes
5answers
391 views
What's the difference between arccos(x) and sec(x)
My question might sound dumb, but I don't really see why the graphics of arccos(x) and sec(x) are different, because as far as I know arccos is the inverse cosine function (cos(x)^-1) and sec equals ...
9
votes
1answer
478 views
Adjoint functors as “conceptual inverses”
The Stanford Encyclopedia of Philosophy's article on category theory claims that adjoint functors can be thought of as "conceptual inverses" of each other.
For example, the forgetful functor "ought ...
9
votes
2answers
188 views
On the convexity of element-wise norm 1 of the inverse
Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as
$$
\|A\|_1= \sum_{i,j} |A_{i,j}|.
$$
Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
8
votes
9answers
884 views
Why is $\frac{1}{\frac{1}{X}}=X$?
Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$
And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
8
votes
6answers
625 views
If $A^2$ is invertible, then $A$ is also invertible?
True or False: If $A^2$ is invertible, then $A$ is also invertible.
($A$ is a matrix here.)
The answer is true. I was trying to come up with an example that makes this false.
But I couldn't. ...
8
votes
5answers
272 views
Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist? Why?
Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist, given that $\mathbf A$ is a square and positive (and possibly singular) matrix and $\epsilon$ is a small positive number? I want to use this to ...
7
votes
1answer
123 views
Inverse of a block matrix
I have a special case where $X=\left(\begin{array}{cc}
A & B\\
C & 0
\end{array}\right)$
and:
$X$ is non-singular
$A$ is singular
$B$ is full column rank
$C$ is full row rank
How do you ...
7
votes
1answer
229 views
$\operatorname{arsinh}$ vs $\operatorname{arcsinh}$
I note that some people like to write the inverse hyperbolic functions not with the prefix "arc" (like regular inverse trigonometric functions), but rather "ar". This is because the prefix "arc" (for ...
6
votes
4answers
181 views
Proof: if the graphs of $y=f(x)$ and $y=f^{-1}(x)$ intersect, they do so on the line $y=x$
This came out of a textbook problem, and as Lubin pointed out below, it's not actually true as originally stated. I'm guessing it should be restated as: If the graphs of $y=f(x)$ and $y=f^{-1}(x)$ ...
6
votes
1answer
80 views
My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible
Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial.
Here's my ...
5
votes
6answers
275 views
What is the inverse function of $\ x^2+x$?
I think the title says it all; I'm looking for the inverse function of $\ x^2+x$, and I have no idea how to do it. I thought maybe you could use the quadratic equation or something. I would be ...
5
votes
3answers
293 views
If $A$ and$ I+AB$ are invertible, show $I+BA$ is also invertible
Show that if $A$ and $I+AB$ are invertible, then $I+BA$ is also invertible with
$$(I+BA)^{-1} = A^{-1}(I+AB)^{-1}A$$
5
votes
2answers
107 views
Help finding inverse of $f(x)=\frac{e^x-e^{-x}}{2}$
I'm trying to find the inverse of $f(x)=\frac{e^x-e^{-x}}{2}$. My textbook says $f^{-1}(x)=\ln(x+\sqrt{x^2+1})$, but I haven't been able to get that answer. Switching $x$ and $y$, I tried solving for ...
5
votes
2answers
63 views
Inverse and derivative of a function [duplicate]
Find an example of an inverse function f(x) such that its derivative is the same as its inverse.
I tried many different functions but non of them worked.
5
votes
3answers
278 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 ...
5
votes
1answer
102 views
Is there a name for this type of inverse?
I have a function $f : A \to B$ and an inverse $f^{-1} : B \to A$, and the only property of the inverse is that $(f \circ f^{-1} \circ f)(x) = f(x)$. In particular, it is not necessarily true that ...
5
votes
2answers
97 views
How to formally show that $f(z)$ is analytic at $z=0$?
Let $z$ be a complex number. Let $$f(z)=\dfrac{1}{\frac{1}{z}+\ln(\frac{1}{z})}.$$ How to formally show that $f(z)$ is analytic at $z=0$?
I know that for small $z$ we have ...
5
votes
1answer
27 views
Finding the inverse of a map from $CP^1$ to $S^2$
Given the map:
$$f:CP^1 \to S^2\ ,\ f[z:w] = \left(\frac{2\mbox{Re}(w\bar{z})}{|w|^2+|z|^2},\frac{2\mbox{Im}(w\bar{z})}{|w|^2+|z|^2}, \frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$
How would I go about ...
5
votes
1answer
217 views
Inverse function of $\operatorname{li}(x)$ over $x>\mu$?
How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$?
Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...
5
votes
0answers
219 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 ...
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} * ...
4
votes
2answers
211 views
How long does it take to consume the same amount of food
For a group of 32 students food lasts for 45 days. For how many days will the same food last for 72 students?
4
votes
4answers
175 views
How to find inverse of the function $f(x)=\sin(x)\ln(x)$
My friend asked me to solve it, but I can't.
If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$?
I have no idea how to find the solution. I try to find
...
4
votes
3answers
519 views
How to invert this exponential function to solve for x: $y = a \exp(bx) + c \exp(dx)$?
Cheers.
So if I don't make sense, I have a value for $y$, I need to know what $x$ is.
$$y = a \exp(bx) + c \exp(dx)$$
$a = 12.85$,
$b = 0.001857$,
$c = -54.24$,
$d = -0.05316$
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 ...
4
votes
3answers
90 views
Matrix Inverses
So in class we have been discussing matrix inverses and the quickest way that I know of is to get a matrix A, and put it side by side with the identity matrix, like $[A|I_{n}]$ and apply the ...
4
votes
2answers
131 views
Solve equation $\tfrac 1x (e^x-1) = \alpha$
I have the equation $\tfrac 1x (e^x-1) = \alpha$ for an positive $\alpha \in \mathbb{R}^+$ which I want to solve for $x\in \mathbb R$ (most of all I am interested in the solution $x > 0$ for ...
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 ...
4
votes
2answers
304 views
Inverse of symmetric matrix $M = A A^\top$
I have a matrix, generated by the product of a non-square matrix with its own transpose:
$$M = A A^\top.$$
I need the inverse of $M$, assuming $\det(M) \neq 0$.
Given the nature of the matrix $M$, ...
4
votes
3answers
83 views
Inverses where $f(g(x))=x\neq g(f(x))$
If $f: \mathbb{Z\to Z}$ and $G: \mathbb{Z\to Z}$, find $f$, and $g$ such that $f(g(x))=x\neq g(f(x))$.
I can find lots of $f$ and $g$ that aren't equal when composed with each other, but I have no ...
4
votes
1answer
118 views
If $z$ is the unique element of a monoid such that $uzu=u$, is $u$ invertible?
This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success.
Let $M$ be a monoid and $u\in M$. ...
4
votes
2answers
85 views
Diffeomorphism from Inverse function theorem
I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
4
votes
1answer
76 views
Inverse of matrices with 3 parts!
I just wonder if there is any closed form solution for the inverse of matrices with following form, or if it's possible to decompose them.
$
\left[\begin{array}{cccccccccc}
{\color{red}1} & ...
4
votes
1answer
80 views
How do we know how many branches the inverse function of an elementary function has?
How do we know how many branches the inverse function of an elementary function has ?
For instance Lambert W function. How do we know how many branches it has at e.g. $z=-0.5$ , $z=0$ , $z=0.5$ or ...
4
votes
1answer
85 views
solve $ y = (A+B^{-1})x $ for $x$
I wish to solve numerically for $x$,
$$ y = (A+B^{-1})x $$
with $A, B$ positive definite. So,
$$ x = (A+B^{-1})^{-1}y $$
I would like to avoid calculating $B^{-1}$ since that's generally bad.
...
4
votes
2answers
34 views
norm of inverse less than 1
I just wanna ask if what I am doing here make sense:
Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
4
votes
1answer
185 views
Super logarithmic inverse of tetration
What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$?
Is it $slog^{x}_{2}$?
4
votes
0answers
53 views
Show that there exist a real number $a≠0$, such that the fiber $f^{-1}(a)$ is a finite set
Let $f:ℝ→ℝ$ be a real analytic function. The function $f$ is given by:
$$f(s)=N^{s/2}(2π)^{-s}Γ(s)∑_{n=1}^{∞}a_{n}/n^{s}$$
where $a_{n}$ are the coefficients of a Dirichlet series, $N$ is a natural ...
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 ...
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 ...
4
votes
0answers
60 views
Explicit quasi-inverse of Künneth-isomorphism?
With $A_X$ the complex of $\mathbb{R}$-differential forms on $X$, the Künneth theorem states that
\begin{align*}
A_X \otimes A_Y &\to A_{X \times Y}, \\
(\omega,\eta) &\mapsto {\rm ...
3
votes
2answers
440 views
What does $M^{-1}RM$ represent?
I'm a bit confused about the use of $M^{-1}RM$ where $R$ is a transformation matrix.
Actually I was looking at the script here which reads and renders bvh files. But, I could not understand the ...
3
votes
3answers
194 views
How to invert this function? (Inverse exponential function with arctan)
How to invert this function?
$$
y = e^{\arctan(x^5)}
$$
3
votes
3answers
108 views
Find the inverse of a $4\times4$ matrix
My matrix looks like this:
$$\left(\begin{array}{rrrr}
1& 1 & 1 & 1\\
1& -1 & 1 & 0\\
1& 1 & 0 & 0\\
1& 0 & 0 & 0
...
3
votes
4answers
77 views
Is there a good intuitive way to understand why matrix B is inverse of A when matrix A|I is turned into I|B
I'm looking for some help with my intuition of basic matrix operations, specifically finding a matrix's inverse (as per my subject line). I have no problems with the steps. The basic row operations ...
3
votes
4answers
109 views
How do you take the multiplicative inverse of a p-adic number?
I am reading the wiki page for p-adic numbers and it states that they are a field extension of the rationals so each member has to have a modular multiplicative inverse.
So how would I take the ...
3
votes
2answers
379 views
An ill-conditioned matrix
If C is an ill-conditioned matrix and I want to get the inverse, one way is to take a pseudo-inverse of some sort. Instead, is the following, which uses the (normal) inverse, also a way to deal with ...
3
votes
2answers
63 views
Given the product of a unitary matrix and an orthogonal matrix, can it be easily inverted _without_ knowing these factors?
Given the product $M$ of a unitary matrix $U$ (i.e. $U^\dagger U=1$) and an orthogonal matrix $O$ (i.e. $O^TO=1$), can it be easily inverted without knowing $U$ and $O$?
Sure enough, if $M=UO$, then ...
3
votes
4answers
128 views
How to show $AB^{-1}A=A$
Let $$A^{n \times n}=\begin{pmatrix} a & b &b & \dots & b \\ b & a &b & \dots & b \\ b & b & a & \dots & b \\ \vdots & \vdots & \vdots & ...
3
votes
1answer
63 views
Find whether or not an inverse exists algebraically
Is there an algebraic(without graphs) way to determine the existence of a function's inverse without using calculus?
I'm an undergrad engineer and can obviously solve this using basic calculus, but ...
