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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The differentiability class of the inverse function

Here's the final part of a proof (from Marden's Elementary Classical Analysis) of the inverse function theorem, where we have been given that $f$ is of class $C^p$: Could someone please explain the ...
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Proof for diagonalizable matrix

Let $A \in M_n(\mathbb C)$ be invertible. Prove that $A$ is diagonalizable if and only if $A^{-1}$ is diagonalizable. This is what I have for one direction of the proof: Suppose $A$ is ...
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107 views

Finding the multiplicative inverse of an element in $\mathbb Q[x]/(x^3-2)$

I have a problem here that asks: "Express the multiplicative inverse of $1+2^{1/3}-3\cdot2^{2/3}$ as $a_0+a_1\cdot2^{1/3}+a_2\cdot2^{2/3}$." I believe they are asking us to find it by utilizing the ...
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1answer
52 views

How would I show this bijection and also calculate its inverse of the function f?

I want to show that f(x) is bijective and calculate it's inverse. Let f (x) : R → R be defined by f (x) = (3x/5) + 7 I understand that a bijection must be injective and surjective but ...
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1answer
48 views

Inverse CDF of a Standard Normal Variable

In many applications, for example Monte Carlo methods, we require the inverse CDF of a standard normal random variable. But the CDF: $$ \Phi(x)= \int_{-\infty}^ x \frac{1}{\sqrt{2\pi}} e^{-t^2/2} dt ...
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Continuous function that is invertible in one argument---is its inverse continuous in both arguments?

Suppose that $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous function and that it is invertible in its second argument, i.e. for every $x \in \mathbb{R}$, $f(x,\cdot)$ is invertible with ...
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3answers
130 views

Lower triangular matrices [duplicate]

Is the inverse of an invertible and lower triangular matrix still both lower triangular and invertible?
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67 views

An invertible matrix

Given Matrix $A$, checking that its diagonal elements are nonzero or whether its determinant is nonzero, can we say the matrix is invertible for sure? Are there other properties that by looking at the ...
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53 views

Iterative update of pseudo inverse solution

I have an overdetermined linear problem of the form $A x = b$, which is solved in least squares sense using the Moore–Penrose pseudo invers. The issue now is, that over time additional constraints and ...
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1answer
26 views

Finding the correct angle from inverse cosine?

For my math homework, I have to find an angle of rotation, $\theta$, by cos $\theta$ = $-\sqrt3/2$. When I plug this into my calculator, I get 5$\pi$/6, but the correct answer is -5$\pi$/6. What is ...
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25 views

Inverse of a matrix expression after SVD

Let: $UDV^T$ be the SVD decomposition of $A$; $\lambda\in\mathbb{R}$ and $I_n$ the identity matrix Why is the following true? $(VD^2V^T+\lambda I_n)^{-1} = V(D^2+\lambda I_n)^{-1}V^T$
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Least Square with homogeneous solution!

I've read somewhere that: $x=A^+b+(I-A^+A)Z$ is a solution for $Ax=b$ ,when is doesn't have a particular solution. where $A^+$ indicates the pseudo-inverse and $Z$ is an arbitrary vector!!! I know ...
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34 views

Show that this is one to one continuous and find its inverse which is continuous as well.

Let's define $\phi: \Bbb R^2 \to S$ for $S$ is subset of $\Bbb R^3$ For constant $a,b,c,d$ and $c\not =0$ $$\phi(x,y)=(x,y, \frac{d-ax-by}{c})$$ I want to show that the function $\phi$ is 1-1 ...
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1answer
40 views

Stuck at Extended Euclidean Algorithm to solve equation

I'm trying to solve the following function via the Extended Euclidean Algorithm, but I'm stuck at the last step where I need to sub in sub 2. ...
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49 views

Example of sets $A$ and $B$ and functions $F$ and $G$ such that $F: A \rightarrow B, G: B \rightarrow A, G \circ F = I_{A}$, and $G \neq F^{-1}$

Give an example of sets $A$ and $B$ and functions $F$ and $G$ such that $F: A \rightarrow B, G: B \rightarrow A, G \circ F = I_{A}$, and $G \neq F^{-1}$ I was thinking maybe $F$ can be a function ...
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Inverse of a 4x4 matrix with variables

I missed my class on the inverses of matrices. I'm catching up well, but there's a problem in the book that got me stumped. It's a 4x4 matrix that is almost an identity matrix, but the bottom row ...
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1answer
25 views

Simplify functions involving modular arithmetic

In this question, the answer says that f o g(x) = x. But I am unable to get this result. The expression I am able to get is that f o g(x) = 7*(x mod 3) + 57*(x mod 7) (mod 21). I am unable to ...
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43 views

Inverse Laplace transform of ratio of polynomials

I would like to understand if the inverse laplace transform of these functions gives something in terms of known functions (such as exponentials..see my examples). ...
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1answer
30 views

Inverse of function arcsin

I'm having trouble finding the solution of the inverse of the function ${\rm f}\left(y\right) = \arcsin\left(\,3 - x2\,\right)$ Isn't $\arcsin$ the inverse of $\sin$ ?. This is what I have now as ...
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1answer
23 views

Parametric Curves Existence of Tangent

If $\frac{dy}{dt}$ and $\frac{dx}{dt}$ exist, then does $\frac{dy}{dx}$ always exist when $\frac{dx}{dt} \not=0$? Indeed, this is a very simple question. Sorry but I'm just a beginner for ...
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How to calculate the inverse of the line integeral.

Let $f$ be a polynomial function, $$ f(x) = a_0 + a_1 x + ... + a_d x^d $$ where $a_0$, $a_1$, ..., $a_d$ are parameters and usually $d \le 6$. Let $g$ be the line integral of $f$, $$ g(x) = ...
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105 views

Finding inverse of a matrix

This question is in my assignment. We are not allowed to use any symbol to represent any elementary row and column operations used in the solution. We must solve it step-by-step. Please help me to ...
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1answer
81 views

Inverse function, power set.

How to prove, that for every function $F: P(\mathbb N) \rightarrow P(\mathbb N)$, where: $F(\mathbb N)=\mathbb N$ $F(\emptyset)=\emptyset$ $F(\bigcup \Xi)=\bigcup\{F(X)|X\in\Xi\}$ for every $\ ...
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1answer
84 views

Inverse of Positive definite matrix

Assume $P$ is a non-negative positive definite matrix. It is well known what $P^{-1}$ is also positive definite and thus all its diagonal entries are positive. Can we say something about the off ...
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Calculating $\text{erf}^{-1}(z)$ for $z\in\mathbb{C}$

All the information I found about inverse error function $\text{erf}^{-1}(z)$ was about $z\in\mathbb{R}$. Also I found some Taylor expansions for it, but as the function is unbounded near $z=\pm1$, ...
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40 views

Injective and Surjective Functions on Sets

I'm fairly new to math proofs. I've been looking for some counterexamples to the following theorems, especially the second one. I haven't been able to think of a scenario. Are the following theorems ...
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45 views

Finding the Inverse Laplace transform using the Step and Shift theorems

I want to find the Inverse Laplace Transformation of the function given above. I used the step and shift theorems to come up with an answer. Can someone simply verify the answer. This is my first ...
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211 views

From matrices to bipartite graphs

Assume $G(A,B)$ is a bipartite graph and assume $L(G)$ is the adjacency matrix of its line graph. define $$B=[3\text{I}+L(G)]^{-1}$$. Is it always the case that for each edge $e=(a,b)\in G$, we have: ...
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326 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 ...
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A square matrix A is invertible if and only if det A ≠ 0. Use the theorem above to find all values of k for which A is invertible

$$\begin{pmatrix} k & k & 0 \\ k^2 & 25 & k^2 \\ 0 & k & k \end{pmatrix}?$$ I did a sample question before this one: $$\begin{pmatrix} k & k & 0 \\ k^2 & 16 & ...
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Tricky question in Matrices! [closed]

Define $$A=[\text{I}+\sum_{k=1}^{m}u_{k}u_{k}^T]^{-1}$$, where for each $u_k$ is a $0-1$ column vector. Prove that for every $1\leq k \leq m$ $$Au_{k}u_{k}^T\geq0$$ i.e. each entry of $Au_ku_k^T$ ...
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56 views

Laplace Transformation spring question

Here is the question: http://i.imgur.com/XAH2UnX.jpg I can't seem to get the answer. Are those values in the writing like 1N/m even relevant? Can someone give me some direction? Thanks!
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Ultrametric matrices and their inverse

A non-negative square matrix $A$ is ultrametric iff: $A(i,i)>\{A(i,k),A(k,i)\}\forall k,i$ $A(i,j)\geq\min\{A(i,k),A(k,j)\}\forall i,j,k$ It is well-known that the inverse of non-negative ...
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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 ...
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1answer
51 views

Calculating Inverse function

Please help me with the following question: Calculate, if possible, the inverse of the following functions: (i) $f(x) = (2x - 2)^5$ (ii) $f(x) = (2x - 3)/4$ (iii) $f(x) = x^2 + 1,$ for $ x \geq ...
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Calculating inverse functions

Could someone please help me with the following questions? Calculate, if possible, the inverse of the functions: (i) $f(x) = (2x - 2)^5$; (ii) $f(x) = \dfrac{(2x - 3)}{4}$; (iii) $f(x) = x^2 + ...
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29 views

Conversion of roots of a polynomial

I'm wondering, given a polynomial $P(x)$ with roots $r_i (1\le i\le n)$, how to determine the polynomial $Q(x)$ such that its roots are $r'_i=f(r_i)$. For example, if $P(x)=x^2-x-6=(x-3)(x+2)$ and ...
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101 views

If $g(x)=3+x+e^x$, then how do I find $g^{-1}(4)$?

If $g(x)=3+x+e^x$, then how do I find $g^{-1}(4)$? I took $g(x)=y$ and tried to solve the problem, but i could not get the solution.So, please help me by providing me the solution to my question.
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Can I find the Pseudoinverse (Moore-Penrose inverse) just by knowing the one-sided inverses of a matrix?

Consider a matrix such as $B = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \end{bmatrix}$. I know how to compute the right inverses (or in the case of $m\geq n$ the left inverses) and ...
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1answer
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Trick: Substitution in inverse trigonometry.

My friends say, it is some what difficult to know, which trigonometric function has to be substituted in the inverse trigonometric equations, to get the correct solution. So, I thought to take up this ...
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20 views

Inverse of $\{a_1 A_1,…,a_n A_n\}$

$a_1,...,a_n\in \mathbb{R}$ $A_1,...,A_n$ are the rows of the invertible matrix A I am trying to find a regular formula for this. Is it possible? Thanks for help!
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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 ...
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54 views

Solution of matrix equations

$$A=\begin{bmatrix} 3 & -2 & -1 \\ 1 & 2 & 1 \\ -1 & 1 & 1 \end{bmatrix}, X= \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 1 \\ 7 \\ 2\end{bmatrix}$$ ...
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1answer
84 views

Finding the inverse function

The question is to find the inverse function of $$f(x)=x-(2\sqrt{x})+1$$ I first found that the domain of definition is $\,x\ge 0$ Then studied the variation of the function and it is decreasing ...
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254 views

Showing a function is bijective and finding its inverse

The function f: ℝ2-> ℝ2 is defined by f(x,y)=(2x+3y,x+2y). Show that f is bijective and find its inverse. I've got so far: Bijective = 1-1 and onto. 1-1 if (2x1+3y1,x1+2y1)=(2x2+3y2,x2+2y2) Then ...
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70 views

derivative of product of 2 inverse matrices

I was trying to differentiate the equation below: $$ \frac{\partial a^T X^{-T}X^{-1}a} {\partial X} $$ where X is invertible but not symmetric and $X^{-T}$ means transpose of inverse of X. In the ...
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94 views

Find the inverse z-transform of $E(z)=\frac{z+1}{(z-1)(z-0.6)}$

$$\begin{align} E(z)&=\frac{z+1}{(z-1)(z-0.6)}\\ \frac{z+1}{(z-1)(z-0.6)}&=\frac{A}{(z-1)}+\frac{B}{(z-0.6)}\\ z+1&=A(z-0.6)+B(z-1) \end{align}$$ set z=0.6: $$\begin{align} ...
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1answer
29 views

About the inverse matrix of the form $(I+cH^{-1})^{-1}$.

Given $(I+cH^{-1})^{-1}$, where $c$ is a constant and $H$ is a $\mathbb{R}^{n\times n}$ matrix. Suppose $(I+cH^{-1})^{-1}$ has a inverse matrix. Is there any way to calculate $(I+cH^{-1})^{-1}$ ...
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1answer
31 views

Inverse Transformation

Consider the coordinate transformation $$ \varphi\colon\mathbb{R}^2\to\mathbb{R}^2, (x,y)\mapsto (y-\arctan(x),y+\arctan(x)). $$ To make it more easy, I set: $$ ...
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2answers
137 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 ...