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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40 views

complex and decimal tetration

So recently in the blog post on tetration, it talked about tetration with nice clean powers (calling them these because I don't know the right term). But how does it work when given a complex power? ...
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1answer
41 views

Proving this function is an open map

Prove the function $f(x, y, z) = (x^3, y^2-z^2, yz)$ is an open map from $\mathbb{R^3}$ to $\mathbb{R^3}$ (i.e for every open set $U$ of $\mathbb{R^3}$, $f(U)$ is open). I know, as an application of ...
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1answer
33 views

Solving equations with matrices

Say I have $4$ simultaneous equations \begin{cases} 4.3S_1 - P = T \\ 8S_2 - P = T \\ 5.5S_3 - P = T \\ S_1 + S_2 + S_3 = T. \end{cases} I'm trying to solve these in Excel using MINVERSE and MMULT ...
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1answer
83 views

If $f(AB) =f(A)f(B)$, then $A$ is inversible iff $f(A)\neq 0$

Let $f:\mathscr M_n(\mathbb K) \to \mathbb K$ be a non constant function such as $f(AB) = f(A)f(B)$ for all $A,B$ in $\mathscr M_n(\mathbb K)$. The question is to show that $M\in GL_n(\mathbb K)$ iff ...
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208 views

find x where $x^{11} \mod 41 = 10$

In a previous part of the question, I am asked to find $11^{-1} \mod 40$. I've done that, the answer's $11$. The question continues: find $x$ where $x^{11} \mod 41 = 10$ showing how you could get ...
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3answers
73 views

Solve logarithmic equation for $x$ to find the inverse of $f(x)= \ln(x+\sqrt{x^2+1})$

Let $f(x)= \ln(x+\sqrt{x^2+1})$. Find $f^{-1}(x)$. Here is what I got so far: $y= \ln(x+\sqrt{x^2+1})$, rewrite as $x= \ln(y+\sqrt{y^2+1})$, then $$e^x= y+\sqrt{y^2+1}$$ $$e^x-y= \sqrt{y^2+1}$$ ...
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1answer
119 views

Questions about matrix rank, trace, and invertibility,

(a) Prove that a square matrix $T$ of rank one has $\text{tr}(T)=0$ if and only if $T^2=0$. (b) Consider a matrix $A$ of the form $A=aI+T$, where $a\ne0$, $I$ is the identity matrix, and $T$ has ...
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1answer
84 views

Mistake in integrating the inverse of a function?

The problem I have is: $$\int_0^{\frac1{\sqrt{3}}}\sqrt{x+\sqrt{x^2+1}}dx$$ I'm not asking how to solve it, I'm asking if anyone can point out to me the error that I have made: By differentiating ...
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107 views

Fastest way to find modular multiplicative inverse

I am looking for a fast way to find the modular multiplicate inverse of an integer $a$ in mod $p$. I am mainly interested in ...
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1answer
71 views

inverse of a tridiagonal matrix

Let $${A_{n \times n}} = \left[ {\begin{array}{*{20}{c}} {-2}&{1}&{}&{}&{}\\ {1}&{-2}&{1}&{}&{}\\ {}&{1}&{\ddots}&{\ddots}&{}\\ ...
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2answers
70 views

Finding inverse of a composite function

Let $f (x) = x^{3}+x$ and $g (x) =x^{3} -x$ for all x. I have to find derivative of $g\circ f^{-1}$ at $x=2$. My textbook did this: $(g \circ f^{-1})' (2) = \lim \limits_{h \to 0} \dfrac{g \circ ...
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0answers
12 views

Efficient inverses of many related matrices [duplicate]

Say I have a $N$-by-$N$ positive definite real matrix $\Sigma$ and I wish to compute the inverses (or equivalently Cholesky decompositions) of $(\Sigma + a_k I)^{-1}$ for a set of $K$ positive $a_k$. ...
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1answer
207 views

Finishing a proof: $f$ is injective if and only if it has a left inverse

I've already done a lot of searching (in particular: https://www.proofwiki.org/wiki/Injection_iff_Left_Inverse) to try to prove this statement: $f: A \to B$ is injective if and only if it has a ...
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2answers
68 views

Is the preimage of a bounded set also bounded?

I need to prove the following statement: Let $f:\mathbb{C}\rightarrow\mathbb{C}$ a continuous function and $B \subseteq \mathbb{C}$ bounded, implies, that the set $A=f^{-1}(B)$ to be bounded. I do ...
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1answer
41 views

Finding a differentiable inverse of $f(x)=x/\cos x$

Let $$ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \rightarrow \mathbb{R} $$ be defined by $$ f(x) = \frac{x}{\cos x}. $$ We're supposed to show that $f$ has a differentiable inverse $$f^{(-1)}$$ ...
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2answers
46 views

What's $\int \frac{1}{\sqrt{25-x^2}}$ [duplicate]

What is $$\int \frac{1}{\sqrt{25-x^2}}$$ WolframAlpha says $\sin^{-1}(\frac{x}{5})$ while I got $\frac{1}{5}\sin^{-1}(\frac{x}{5})$. What is correct? Thanks in advance.
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2answers
56 views

Prove that a continuous inverse-transformation of $f: [0,1) \cup \{ 2 \} \to [0,1]$ exists

I am having this transformation $f: [0,1) \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ I've already proved that it is continuous. Question: Is ...
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1answer
29 views

Compute $\left(a_{i}A+B\right)^{-1},\qquad i=1,\ldots,N$ efficiently?

I need to compute the inverse matrix: $$(a_i A+B)^{-1}, \qquad i=1,\ldots,N$$ where $N$ is a large number. $A$ and $B$ are general $M\times M$ matrices independent of $i$. The only thing that ...
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2answers
404 views

Why aren't integration and differentiation inverses of each other?

Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral: $$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int ...
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1answer
28 views

inverse of a point $p$ respect to the circle $|z-z_0 |= r$ in complex

I was solving a problem to find the inverse of a point $p$ respect to the circle $|z-z_0|=r$. In my question I had to find inverse of $1+i$ w.r.t circle $|z+1-2i| = 2$. I applied the formula $q = z_0 ...
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1answer
78 views

Derivative of scalar function with respect to vector

Suppose I have three constant symmetric matrix $\mathbf{M}_{n\times n}$, $\mathbf{C}_{n\times n}$ and $\mathbf{D}_{n\times n}$ and two variable vectors $\mathbf{q}_{n\times 1}$ and ...
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1answer
66 views

Finding the inverse of a recursive function?

Let's say I have this function $$f(x) = \sum_{i=0}^{x-1}f(i)$$ provided $f(0) = 0, f(1) = 1$ and $x \in \mathbb Z$. This function is evidently one-to-one on $[3, \infty) $. Is there an inverse to this ...
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1answer
58 views

The invertibility of matrix $(I - XX')$?

$I$ is an identity matrix of size $n \times n$. $X$ is a matrix of size $n \times k$(Assuming $k \leq n$). As we know, $(I+XX')$ is invertible. Because $(I+XX') = (I(blank)X)*(I(blank)X)'$, where $(I ...
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1answer
68 views

Definition of inverse function

I have been wondering... Is there a mathematical equation for the inverse of a function? I mean apart from the typical "replace the x's with y's" way... I tried using the inverse function derivative ...
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1answer
60 views

How to simplify the inverse $(AB)^{-1}$ for rectangular $A$ and $B$?

Assume real rectangular matrices $A$ and $B$, where $A$ is $m \times n$, $B$ is $n \times m$, $m<n$, and the $m \times m$ product $AB$ is invertible. What are some possible strategies for ...
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44 views

Derivative Inverse of a function

I have a question: $\begin{array}{lrl} \mbox{If :} & f(x) & = x^5 + 3x^3 + 2x + 1 \\ \mbox{And :} & g(x) & = f^{-1} (x) \\ \mbox{What is :} & g'(7)&\mbox{?} \\ \mbox{What I ...
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51 views

How do you reverse $\frac{100n(n+1)}{2}=c$ to find n given c?

I'm developing a game where the character experience needed by level is given by Gauss' formula multiplied by 100: $ \dfrac{100\mathrm{level}(\mathrm {level}+1)}{2}$. So the experience table is ...
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0answers
19 views

Why is this Quadratic Form Independent of its Parameter in the Limit?

For $\alpha = e^{1/N}$, I have the following upper-triangular $\left(N+1\right)\times\left(N+1\right)$-Toeplitz matrix: $$\tilde{G}^{\left(N\right)}=\begin{pmatrix} 1/2 & 0 & 0 & \cdots ...
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1answer
15 views

$(T^{-1})^i == (T^i)^{-1}$?

I wonder if the hypothesis in the title is true. And if so, some ideas to prove it. I know $(A^T)^{-1} = (A^{-1})^T$ EDIT: Edited the title to match the generic answer. T does not have to be ...
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1answer
22 views

Prove that scalar functions of vectors cannot be inverted

The following seems obvious to me (because information is clearly lost), but I have no idea how to prove it: Suppose we have some arbitrary complex vector $\mathbf{A}$ with $m$ components. Let ...
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1answer
52 views

What is the correct $\det(A^{-1})$

Ok so I think I know why this is incorrect, because of the following: $$\det\frac{1}{ad-bc}\begin{bmatrix} d & -b\\ -c & a \end{bmatrix}\neq \frac{ad-bc}{ad-bc}$$ However, by adding a det ...
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94 views

If $A^2 = O$, is $A = O$?

I think the answer is "no", but I'm trying to find the flaw in this reasoning: $A^2 = O \implies AA = O \implies A^{-1}AA = A^{-1}O \implies A = O$ This shouldn't be true, as far as I know, so what ...
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2answers
32 views

Testing if a function has an inverse.

I was just wondering how you apply the rule: $$f(x_1) = f(x_2) => x_1 = x_2 $$ on the function: $$f(x) = x^3 - 9x^2 +33x +45$$ Any suggestions on how to proceed would be appreciated. EDIT: Yes ...
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1answer
79 views

Vertex Equation of an inverse quadratic function.

I'm working on a graphing web tool using JSXGraph, The user should be able to draw different functions. I was able to allow the user to draw quadratic functions by creating the vertex of the function ...
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3answers
29 views

Orthogonal matrices show that the product is also orthogonal

Show that if A and B are two orthogonal n × n matrices, then so is AB I know orthogonal is when the transpose of the matrix is equal to it's inverse. Please help
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1answer
33 views

For $A, B$ invertible matrices of the same order, is it true that $(A^T + B^T)^{-1} = (A^T)^{-1} + (B^T)^{-1}$?

If $A$ and $B$ are invertible matrices of the same order then is this statement true? Why? $$(A^T + B^T)^{-1} = (A^T)^{-1} + (B^T)^{-1}$$
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1answer
24 views

What kind of a matrix has a unitary diagonalizing matrix?

Suppose $D = P^{-1} A P$. When is $P$ unitary? In other words, what kind of a matrix $A$ should be, such that $D=P^{\dagger}AP$? i.e. what are the conditions a matrix must have to be able to ...
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23 views

Prove proposition on real numbers and inverses.

Prove the following proposition Let $x, y \in \mathbb{ R}>0$. If $x < y$ then $0 < y^{-1 }< x^{-1}.$ So far I've gotten that since $x, y > 0$ then $x^{-1}, y^{-1} > 0$.
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1answer
70 views

Linear Algebra - Find inverse of $A$

I have this problem : $$A = \left(\begin{array}{ccc} 3 & -1 & 1 \\ 2 & 0 & 1 \\ 1 & -1 & 2 \end{array}\right) $$ 1) Show that $A^3-5A^2+8A-4I=0$. 2) Using (1) To find ...
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51 views

What tools should be used to prove that a real function is one-to-one and onto?

Let $A = \mathbb R \setminus \{−1/2\}$ and $B =\mathbb R \setminus \{2\}$. Define $f : A \to B$ by the rule $$f(x) = \frac{4x − 3}{2x+1}$$ for all $x \in A$. Show that $f$ is one to one and onto. Find ...
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18 views

Solving simultaneous equations with matrices

I have a Matrix $B = \begin{pmatrix}2&1\\3&5\end{pmatrix}$ and its inverse $B^{-1}=\frac17\begin{pmatrix}5&-1\\-3&2\end{pmatrix}$ I also have a set of simultaneous equations: ...
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1answer
27 views

What properties do I have if I know $f$ and $f^{-1}$inverse are differentiable?

My goal is to show that $(f^{-1})'(y) = 1/[f'(f^{-1}(y)]$ for all $y$ in $(a,b)$. I have no idea where to start. I know that $f^{-1}$ and $f$ are differentiable.
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1answer
132 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
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7answers
146 views

$f \circ g =\operatorname{ id}$ and $g \circ f \neq \operatorname{id}$?

Give two functions $f$ and $g$ s.t. $$f \circ g = \operatorname{id}$$ but $$g \circ f \neq \operatorname{id}$$ or a proof that this is impossible. This must be trivial, but I can't figure it out :) ...
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2answers
35 views

Showing that $\mathcal{G}(\ell_2)$ is not dense in $\mathcal{B}(\ell_2)$ via the right shift

This is my question: Is $\mathcal{G}(\ell_2)$ is dense in $\mathcal{B}(\ell_2)$? I am attempting to show that it is not by showing that the right-shift - call it $T:\ell_2 \rightarrow \ell_2$ - ...
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44 views

Inverse Relation of Irreflexive Property.

We are taking the inverse of relation to check that inverse of R is transitive, reflexive , symmetric and anti-symmetric to as it is on R (not inverse).. My question is that why we are not taking the ...
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46 views

Abscissa of absolute convergence of a Dirichlet series

I'd like some help to prove the following theorem : Let $\sum_{n \geq 1}\frac{f(n)}{n^s}$ and $\sum_{n \geq 1}\frac{g(n)}{n^s}$ be two Dirichlet series with respective abscissas of absolute ...
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1answer
54 views

Is this notation for inverse functions bad?

I'm trying to find useful notation for inverse functions that isn't too much in conflict with other notation already in use, but I'm wondering if this notation will come back and bite me in the ...
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0answers
21 views

Compute new inverse when old inverse and new and old matrix known

Say I have a matrix $M$ and know its inverse $M^{-1}$. Then every element changes so that $M'=M+(M'-M)$. Is there a fast way to find $M'^{-1}$ from this information? That is without computing the new ...
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1answer
33 views

The relation of domain and image of a function and its inverse

Theorem: Let both $f$ and $f^{-1}$ be functions. $\newcommand{\dom}{\operatorname{dom}}\newcommand{\im}{\operatorname{im}}$ Then $\dom(f) = \im(f^{-1})$ and $\dom(f^{-1}) = \im(f)$. Let $f: X ...