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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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
3k views

Inverse of upper triangular matrix

I have an upper triangular matrix that I want to solve the inverse for. I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity ...
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1answer
31 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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0answers
105 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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2answers
44 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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45 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
15 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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1answer
36 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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0answers
16 views

Inverse transformation of continous transformation is bounded

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with $B \subseteq \mathbb C $ bounded. Now I want to proove that $ A = f^{-1} (B)$ is bounded! How can I proove that this ...
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1answer
51 views

Multiplicative Inverse of a Power Series

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = ...
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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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3answers
60 views

Inverse of a Function exists iff Function is bijective

How to mathematically prove that inverse of a function, let's say, $f^{-1}$, exists, if and only if $f$ is bijective? I know how to prove it using diagrams but I'm looking for a rather mathematical ...
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1answer
19 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
50 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
44 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
701 views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
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1answer
23 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

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
42 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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2answers
33 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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3answers
46 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
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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
13 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
18 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
49 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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1answer
70 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
31 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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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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29 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
281 views

Inverse Trig Functions with Double Angle Formulas

I am studying for a quiz tomorrow and one of the sections I am studying involves rewriting quantities as algebraic expressions of $x$. One of the problems I am having trouble with is: $$\sin ...
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1answer
21 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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1answer
49 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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Is the pseudoinverse of a singular, lower triangular matrix itself lower triangular?

Suppose $L\in\mathbb{R}^{n\times n}$ is a singular, lower triangular matrix. Is its psuedoinverse, $L^\dagger\in\mathbb{n\times n}$, also lower triangular? I have already proved by induction that the ...
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88 views

Some questions about the pseudoinverse of a matrix

For every mxn-matrix A with real entries, there exist a unique nxm-matrix B, also with real entries, such that $$ABA = A$$ $$BAB = B$$ $$AB = (AB)^T$$ $$BA = (BA)^T$$ B is called the pseudoinverse ...
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1answer
1k views

Blockwise Moore-Penrose pseudoinverse?

There exists a convenient formula for computing the inverse of a block matrix consisting of 4 matrices $\mathbf{A, B, C, D}$ $ \begin{bmatrix}\mathbf{A} & \mathbf{B} \\ \mathbf{C} & ...
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1answer
37 views

Pseudoinverse and orthogonal projection

Given the matrix $A= \begin {pmatrix} 1 & 1 &1 \\ -1 & 1 & 0 \\ 0 & 2 &1 \end{pmatrix}$. (i) Determine the orthogonal projection $p:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ on ...
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1answer
89 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 ...
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1answer
44 views

What is the Moore-Penrose pseudoinverse for a hermitian block-matrix with one zero block?

Given a block matrix of the form \begin{pmatrix} A & B^* \\ B & 0 \end{pmatrix} where $A$ is singular (otherwise one could simply use the well-known block matrix inverse), is there a ...
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21 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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62 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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14 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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52 views

True or false? Prove it.

If $A$ is an $n\times n$ invertible matrix and $B$ is an $n\times m$ matrix, then $\operatorname{rank}(AB) = \operatorname{rank}(B)$. Is this true or false? I've tried proven that if $B=0$, then ...
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1answer
24 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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7answers
133 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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22 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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4answers
54 views

Set of all matrices with determinant 0, non-zero

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes ...
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0answers
30 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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26 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
46 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 ...
0
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2answers
317 views

Inverse functions and tangent line

Let $f(x) = \frac14x^3 + 12x + 6$ and let $y = f^{-1}(x)$ be the inverse function of $f$. Determine the $x$-coordinates of the two points on the graph of the inverse function where the tangent line is ...