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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3answers
407 views

How to derive compositions of trigonometric and inverse trigonometric functions?

To prove: $$\sin({\arccos{x}})=\sqrt{1-x^2}$$ $$\cos{\arcsin{x}}=\sqrt{1-x^2}$$ $$\sin{\arctan{x}}=\frac{x}{\sqrt{1+x^2}}$$ $$\cos{\arctan{x}}=\frac{1}{\sqrt{1+x^2}}$$ ...
9
votes
2answers
17k views

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!
3
votes
0answers
399 views

Left inverse iff injective; right inverse iff surjective

For a function $f:A\to B$, the function $g:B\to A$ is called: a left inverse for $f$ if $g\circ f$ is the identity on $A$ (i.e., $g\circ f = {\rm id}_A$); and a right inverse for $f$ if ...
4
votes
2answers
329 views

Right Inverse for Surjective Function

Prove that if $f:X\to Y$ is a surjective function between sets, then there must exist a function $g:Y\rightarrow X$ such that $f\circ g=1_Y$. I know that the identity function is onto, and if $f$ ...
15
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1answer
651 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 ...
6
votes
1answer
2k views

Is the trace of inverse matrix convex?

Hi I would like to know whether the trace of the inverse of a symmetric positive definite matrix $\mathrm{trace}(S^{-1})$ is convex. Actually I know that the trace of a symmetric positive definite ...
5
votes
1answer
395 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 ...
1
vote
1answer
126 views

The inverse of the matrix $\{1/(i+j-1)\}$

Let $n$ be a positive integer. Show that the matrix $$\begin{pmatrix} 1 & 1/2 & 1/3 & \cdots & 1/n \\ 1/2 & 1/3 & 1/4 & \cdots & 1/(n+1) \\ \vdots & \vdots & ...
17
votes
12answers
3k 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 ...
5
votes
3answers
382 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
3answers
703 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 $M$, ...
1
vote
2answers
151 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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vote
1answer
210 views

Finding the inverse of the sum of two symmetric matrices A+B

Consider calculating the inverse of matrix sum $$A+B$$ where A is a symmetric dense matrix while B is a symmetric block diagonal matrix. I am interested in finding an efficient approach to update ...
0
votes
2answers
53 views

Find the range of arcsin$((1-x^2)^{0.5})$

Title says it all, how do you get the answer to this? So far I only reach $0<1-x^2<pi/2$ but I get an invalid answer from here. the correct answer is $0<x<pi/2$. Any help is appreciated, ...
8
votes
1answer
337 views

Simple to state yet tricky question

Define $$A=\left[\mathrm I+\sum_{k=1}^{m_1}v_k v_k^T+\sum_{k=1}^{m_2}u_k u_k^T\right]^{-1},$$ where each $u_k$ and $v_k$ is a $0$-$1$ column vector, and for each $1\leq i \leq n$, the $i$th component ...
5
votes
2answers
144 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 ...
4
votes
1answer
520 views

How to invert a very regular banded Toeplitz matrix?

What's the best way to invert a simple Toeplitz matrix of the following form? $$ A = \begin{bmatrix} 1 & a & 0 & \ldots & \ldots & 0 \\\ a & 1 & a & \ddots & ...
11
votes
5answers
3k 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 ...
8
votes
1answer
869 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 ...
3
votes
1answer
234 views

Inverse of $(A + B)$ and $(A + BCD)$?

Consider $A$ as an arbitrary matrix and $B$ as a symmetric matrix. Since $B$ is symmetric, therefore, it can be written as a $\Gamma \Delta \Gamma'$, where $\Delta$ is a diagonal matrix with ...
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2answers
270 views

Show that A is invertible and that it is Lower Triangular.

Does anybody have a solution to the given word problem below? Let A be a lower triangular n x n matrix with nonzero entries on the diagonal. Show that A is invertible and and that A-inverse is lower ...
13
votes
4answers
874 views

Is $A + A^{-1}$ always invertible?

Let $A$ be an invertible matrix. Then is $A + A^{-1}$ invertible for any $A$? I have a hunch that it's false, but can't really find a way to prove it. If you give a counterexample, could you please ...
8
votes
1answer
210 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 $X$: ...
5
votes
6answers
316 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 ...
4
votes
3answers
2k views

Proving that a right (or left) inverse of a square matrix is unique using only basic matrix operations

Proving that a right (or left) inverse of a square matrix is unique using only basic matrix operations -- i.e. without any reference to higher-order matters like rank, vector spaces or whatever ( ...
3
votes
3answers
174 views

Is $\sqrt{x^2}$ always $\pm x?$

I am wondering if this holds in every single case: $$\sqrt{x^2} = \pm x$$ Specifically in this case: $$\sqrt{\left(\frac{1}{4}\right)^2}$$ In this one we know that the number is positive before ...
2
votes
2answers
846 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 $\left[\frac{1}{f(x)}\right]' = ...
2
votes
1answer
2k views

Homework Help - AP Calculus - Inverse of Polynomial

I know it is a simple problem but I am having trouble. Here is what I have so far: Let $f(x) = x^5 + 2x^3 + x - 1$ a) Find $f(1)$ and $f'(1)$ I have a) done. $f(1)$ is $3$ and $f'(1)$ is ...
2
votes
2answers
307 views

Why is $x^{1/n}$ continuous?

Why is $x^{1/n}$ continuous for positive $x,n$ where $n$ is an integer? I can't see how it follows from the definition of limit. And I don't see any suitable inequalities so is this an application of ...
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vote
2answers
96 views

how to prove this equality

There are two equalities, $\sinh({\cosh }^{-1}x)=\sqrt { {x }^2-1 }\quad (x>1)$ $\cosh({\sinh}^{-1}y)=\sqrt{1+{y}^2}$ prove this equality please.. how to prove it? i cannot try it.. also, ...
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2answers
2k views

RYB and RGB color space conversion

I am working on a project where I need to convert colors defined in RGB (Red, Green, Blue) color space to RYB (Red Yellow Blue). I managed to solve converting a color from RYB to RGB space based on ...
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vote
3answers
286 views

Is there a general way to solve transcendental equations?

To make things definite, let's narrow them and call transcendental equation of the form $$f(x) = 0$$ where $f$ is a real elementary function in the usual sense. For example $$\cos(\pi x) + x^2 = ...
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votes
1answer
156 views

What's the limit of coefficient ratio for a reciprocating power series?

I have a question about the coefficient in the inverse of the power series. Assume $$ f=1-\sum_{i=1}^{\infty}(ck_i)x^i, $$ where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...
8
votes
1answer
2k 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 ...
5
votes
1answer
50 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 ...
4
votes
3answers
701 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. $$y = a \exp(bx) + c \exp(dx)$$ $a = 12.85$, $b = 0.001857$, $c = -54.24$, $d = -0.05316$
3
votes
1answer
152 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 ...
2
votes
3answers
856 views

left inverse is not equal to right inverse [duplicate]

Is it possible to have a function which has both left and right inverse but they are unequal ? A left inverse means the function should be one-to-one whereas a right inverse means the function should ...
2
votes
1answer
40 views

Error in understanding the theorem about the invertibility of an element(coset) of a quotient ring

There's a theorem in Abstract Algebra which states that: An element of a quotient ring $\mathbb{Z}/\langle n \rangle$ or $\mathbb{Z_n}$ that is a coset $\overline{a}$ is invertible iff $a$ and $n$ ...
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vote
1answer
31 views

How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
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vote
3answers
74 views

Suppose R is an integral domain. Prove that $(a)=(b)$ if and only if $b = ua$ where $u$ is in $R^\times$

I am lost on this one. I'm still new to ring theory, as we're only a couple weeks into the course, but it's already well over my head. I know that $R$ is an integral domain, so the additive and ...
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vote
1answer
195 views

Inverse Laplace Transform for $F(s) = (9s-24)/(s^2-6s+13)$

Find the inverse Laplace transform of $\displaystyle F(s) = \frac{9s-24}{s^2-6s+13}$. I have tried factoring out a $3$ from the top and putting it into the form of $\displaystyle\frac{b}{(s-a)^2+b^2}$ ...
1
vote
2answers
107 views

Example of a linear operator on some vector space with more than one right inverse.

In my preparations for an upcoming exam, I'm working through the past exams available from my department. One of the questions asks for an example of a linear operator on some vector space with more ...
1
vote
2answers
83 views

Moore-Penrose Inverse and Standard Inverse

I have read that the Moore-Penrose inverse $A^+$ of a matrix $A$ is the same as the standard inverse $A^{-1}$ in the case $A$ is a square, invertible matrix. Is there any relation between the ...
1
vote
1answer
175 views

Question related to diagonally dominant matrix

A matrix is said to be positive if each entry in the matrix is positive. If $A$ is real, irreducible, diagonally dominant (or strictly dominant matrix) and has positive diagonal and non-positive ...
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vote
3answers
800 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 ...
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votes
0answers
37 views

Contradiction in inverse Laplace transform problem with Mellin's inverse formula?

Let say we have to solve a given differential equation $$ty''+y'+ty=0$$ $$y(0)=1,\ y'(0)=0$$ (which is Bessel equation with the solution $y=J_0 (t)$, of course) with the Laplace transform. Then we ...
0
votes
3answers
68 views

If Q is a p-Sylow-Group of H there is a p-Sylow-Group P of G with $\phi(P)=Q$ while $\phi:G\rightarrow H$ epimorphism

Let G be a finite group and $\phi: G \rightarrow H$ a group-epimorphism. Proof: If $Q\in Syl_p(H)$ there is a $P\in Syl_p(G)$ with $Q=\phi(P)$.
0
votes
2answers
38 views

How do I calculate the inverse of these matrices?

In learning how to rotate vertices about an arbitrary axis in 3D space, I came across the following matrices, which I need to calculate the inverse of to properly "undo" any rotation caused by them: ...
0
votes
0answers
61 views

Sherman-Morrison formula for rank 1 update

If $A$ is nonsingular and if for a particular $i$ and $j$ there is no way to make $A$ singular by changing $a_{ij}$ (rank-$1$-update), then using the Sherman-Morrison formula, what can we conclude ...