Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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Matrix invertibility proof? [closed]

Can it be proven that $A^\top A$ is invertible given just the fact that: if $A^\top Ay = \theta$ then $Ay = \theta$? Here $y$ is a vector and $\theta$ is the vector zero.
1
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1answer
58 views

inverse of $2\times2$ block matrix

I want to compute the inverse of the $2\times2$ block matrix $$ \left(\begin{array}{cc} A & P\\ P^T & 0\\ \end{array}\right), $$ with $A$ an $n\times{n}$ matrix and $P$ and $n\times{m}$ matrix....
0
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2answers
43 views

Prove that $(G,*)$ is a group.

Let $G$ be a non-empty set, $*:G\times{G}\to{G}$ a binary operation that satisfies: $*$ is associative. Exist $e\in{G}$ such that $a*e=a$ $\forall{a\in{G}}$ For all $a\in{G}$ there is $i(a)...
2
votes
1answer
60 views

What is the derivative of $\mathrm{trace}((S^T S)^{-2})$ w.r.t. $S$

I would like to compute the derivative of $\mathrm{trace}((S^T S)^{-2})$ w.r.t. $S$. I know that $\frac{\partial \mathrm{trace}((S^T S)^{-1})}{\partial S} = -2S(S^T S)^{-2}$ but I have a higher order ...
33
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3answers
2k views

Does the sum of the inverses of the sums of the primes converge?

$$\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n} = \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} ... $$ Where $p_n$ is the $n$th prime number, does $\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n}$ ...
0
votes
3answers
34 views

Inverse Function with Fraction.

I'm having an issue with this problem for solving for inverse function: $$f(x) = \frac{9x + 5}{x + 4}$$ Step 1: f(x) to Y. Then, change "$x$" to "$y$" in all cases. $$f(x) = \frac{9y + 5}{y + 4}$$ ...
1
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1answer
29 views

Inverse of a matrix whose elements are arrays

I have a group of data as in the following figure: $$A=\left[\begin{array}{ccc} [0.9\,\,0.6\,\,0.9\,\,0.2] & [0.4\,\,0.3\,\,0.1\,\,0.1] & [0.1\,\,0.3\,\,0.5\,\,0.6] \\ [0.6\,\,0.7\,\,0.2\,\,0....
1
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0answers
47 views

Explicitly understanding the implicit function theorem

Suppose I have a curve $f$ in $\mathbb{R}^2$, the implicit function theorem guarantees the existence of a smooth local inverse of this function $f$. Question: My question is is there a way to ...
2
votes
2answers
69 views

Proving that $(A^n)^{-1} = (A^{-1})^n$ for invertible matrix $A$.

I have seen a proof of the fact that for an invertible matrix $A$, $A^n$ is also invertible and $$ (A^n)^{-1} = (A^{-1})^n. $$ The proof was by induction and it was mentioned that one has to use ...
2
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2answers
35 views

Strictly Convex Implies Invertible Gradient?

If $f:\mathbb R^n \rightarrow \mathbb R$ is strictly convex and continuously differentiable, does this imply that $\nabla f$ is a one-to-one mapping? To be precise, can we say that $x, y \in \mathbb ...
2
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1answer
10 views

Inverse function for non-linear regression purpouses

The Setting: I want to perform a regression onto data of that follows this shape: \begin{equation} U(x):=\sum_{i=1}^N\, a_ix^ie^{-b_ix} \end{equation} where the $a_i\in \mathbb{R}$ and the $b_i \in (...
0
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2answers
34 views

Multiplication of ACB where A and B are invertible produces a matrix of zeros with an Identity block

Consider m*n matrix C and invertible square matrices A and B with m and n dimentions. If all three matrices belong to a Field F(all elements of each matrix belongs to F), we can show that there is A ...
3
votes
1answer
43 views

What is the mathematical term for an operation that is self reversing? [duplicate]

What is the mathematical term for an operation that is self reversing? For example: Multiplying by -1 1/x In general: f(f(x)) = x
1
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1answer
21 views

What is the Right Context of a Word in Formal Langauge Theory?

I am reading this paper and am unable to understand this notation in section 2.2 - The right context of a word $u$ according to a language $W$ is the language {$u^{-1}w$ | $w \in W$}. The ...
2
votes
1answer
65 views

How do you find the Inverse Laplace transformation for a product of $2$ functions?

If $$\mathscr{L}(y)=\frac{ne^{-pt_0}}{n^2+\omega^2}\left(\frac{1}{p+n}+\frac{n}{p^2+\omega^2}-\frac{p}{p^2+\omega^2}\right)$$ show that $$\bbox[yellow] {y=n\left(\frac{e^{-n(t-t_0)}}{n^2+\omega^2}+\...
0
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1answer
68 views

Inverse CDFs - common points in support

$G_{1}$ and $G_{2}$ are CDFs of some non-degenerate distributions specified on $\mathbb{R}$ and $\widetilde{G}_{1}$, $\widetilde{G}_{2}$ are corresponding generalized inverses ($\widetilde{f}(u)=\inf \...
1
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1answer
32 views

Finding an inverse of a matrix with determinants

(An exercise in the chapter: determinants) Let $$A = \left[ \begin{matrix} I_k & U \\ 0 & I_l \end{matrix} \right] $$ Find the inverse of this matrix Since $A$ is upper triangular ...
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1answer
34 views

Using Cayley hamilton therom to find the value of matrix

This is the problem where I don't understand how the equations get cancelled and the answers comes out to be $A^2+A+I$. When I am doing the same thing there $7A^4$ which is't cancelled.
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1answer
26 views

Operations for LU decomposition

Given $X\in\mathbb{R}^{n\mathrm{x}n}$ is invertible, $p,q\in\mathbb{R}^{n}$, $x\in\mathbb{R}$, and assuming the $LU$ decomposition without pivoting of $X$ is known, I have to show the LU ...
0
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2answers
39 views

Inverse of $y=2x^2-12x+13$

I'm having a problem finding the inverse of $y=2x^2-12x+13$. At the end I get to the following: $$x=3 \pm \frac{\sqrt{40+8y}}{4}$$ As far as I know the answer is suppose to be $x= 3 \pm \frac{\sqrt{...
0
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1answer
30 views

How to find an inverse of this function

So I am aware of the general rules to follow to find an inverse of a function, but it seems like I'd need something different for this one: $$f(x) = -2x^3-7x+5$$ if I try what I'm use it I end up ...
1
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2answers
67 views

Explain $\tan^2(\tan^{-1}(x))$ becoming $x^2$ [closed]

How does $\tan^2(\tan^{-1}(x))$ become $x^2$? EDIT (After the correct answer was provided. Added because of criticism.) I feel that the answer should contain a tan somewhere and not just simply $x^...
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0answers
26 views

Find the Inverse Laplace Transform of the following

I have a Laplace tranform in the form given below $\mathcal{L}_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$ Can some one help me to find ...
1
vote
1answer
78 views

Derivative of Nested Matrix Quadratic Form

I have two real matrices: $\mathbf{A} \in \mathbb{R}^{k \times d}$, $\mathbf{B} \in \mathbb{R}^{d \times d}$, where $k \leq d$. Further $\mathbf{B}$ is symmetric. I also have two vectors $\mathbf{c},\...
0
votes
0answers
17 views

Matlab inverse tangent

I have a system that contain complex conjugate pole pair in his transfer function. If I examine the function with Matlab's bode it works great and I get results I ...
1
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1answer
45 views

Find the original function by using convolution theorem

Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it. \begin{equation}\frac{1}{((...
0
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1answer
55 views

L2 Norm of Inverse of Non-square Matrix Multiplication

Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$. Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/upper bound for $\|(AA^T)^{-1}\|_2$ in terms of $\...
2
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0answers
48 views

Finding the multiplicative inverses in fields

Let's say I have the field $F_{11}$. Why does $2$ have the multiplicative inverse $6$? In some of the examples I have, let's say we are looking $F_5$, why are values up to only $2$ considered? So ...
0
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2answers
23 views

Finding a matrix inverse when an equation involving it is a multiple of the identity matrix

Say you had a matrix $A$, and you did an equation like $A^2 - A$, and proved that it was a multiple of $I$. How could you find $A^{-1}$ in the form $rA + sI$ after proving that? I want to do it ...
6
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4answers
430 views

Let A be a square matrix such that $A^3 = 2I$

Let $A$ be a square matrix such that $A^3 = 2I$ i) Prove that $A - I$ is invertible and find its inverse ii) Prove that $A + 2I$ is invertible and find its inverse iii) Using (i) and (ii) or ...
0
votes
1answer
41 views

What is meant by In-Place Matrix Inversion?

I come across the term "In Place Matrix Inversion" a lot in numerical libraries like NumPy and ND4J. What does it mean ? How is it different from the normal matrix inversion ? What are the advantages ...
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0answers
33 views

Matrix Inverse as Series

I am looking for different representations of the inverse of a matrix as a power series. One obvious candidate is the Von Neumann series which is given $$A^{-1} = \sum_{k=0}^{\infty} (I-A)^k$$ ...
1
vote
2answers
62 views

Solve equation of inverse functions

I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and $f^{-1}_2(x)$, by ...
4
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4answers
333 views

Definition of Inverse in Linear and Abstract Algebra

In a linear algebra text, the following is the definition of the inverse of a matrix An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = I_n$$...
2
votes
1answer
78 views

In which cases are $(f\circ g)(x) = (g\circ f)(x)$?

I have found three cases: 1) If $f$ and $g$ are the same function. 2) If $f$ and $g$ are mutually inverse. 3) If both are polynomials of degree $1$ Maybe there are more.
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1answer
54 views

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
0
votes
2answers
56 views

Find the inverse $\dfrac{x}{\|x\|}$ in $\mathbb{R^2}$

I wish to find the inverse of $\dfrac{x}{\|x\|}$, where $x \in \mathbb{R}^2$ Let's do this. Let $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}}$$ $$y_2 = \dfrac{x_2}{\sqrt{x_1^2+x_2^2}}$$ Then $$y_1 = \...
0
votes
2answers
32 views

How to simplify inverse trigonometric function

How to simplify the following equation: $$\sin(2\arccos(x))$$ I am thinking about: $$\arccos(x) = t$$ Then we have: $$\sin(2t) = 2\sin(t)\cos(t)$$ But then how to proceed?
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1answer
20 views

Generate random variate using inverse transform technique of $ f (x) =a (1+|x-2|)$

I need to generate a random variable with density function: $$ f(x)= \begin{cases} a (1+|x-2|) , & {-1 \le x \le 4} \\ 0, & \text{elsewhere} \end{cases} $$ For that I need to inverse the ...
0
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3answers
27 views

Why does $\frac{1}{6e^{2y}}=\frac{1}{2x-8}$ in this context?

This is the context: I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
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2answers
31 views

Asymptotes of $\arctan (2x)$

My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, ...
0
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1answer
20 views

Geometric progression with reverse order

I have the following problem: Find three positive numbers which have the sum of $70$ and create a Geometric progression ($q>0$, increasing). Their inverse sum equals to $4/70$. Thank you!
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2answers
54 views

Is a factorable polynomial invertible?

The reason there exists no quintic formula that finds the roots of a quintic polynomial is simply because some quintic polynomials are irreducible. But reducible quintic polynomials may be invertible ...
2
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3answers
60 views

Can you inverse a funcion by rotating it?

In school i sometimes run on some excercises where you need to calculate something that has an inverse function in it but you cannot find the inverse and you need to work your way around it. I know ...
2
votes
1answer
41 views

Inverse of the composition of two functions

If I have a composition of two functions: $$y = f(g(x),h(x))$$ where both $g(x)$ and $h(x)$ are readily invertible, can I find the inverse of the composition? i.e.: Can I find $x = f^{-1}(y)$? I ...
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1answer
20 views

Invariant under $x \rightarrow 1/x$?

I started thinking on the following problem. I am interested in finding complex functions of a complex variable such that $\phi(z)=\phi(z^{-1})$ So far, all I could come up with was a family of ...
4
votes
1answer
31 views

Does $\sin^{-1}x$ has a vertical tangent

I read that the function $f(x)$ has a vertical tangent at $x=a$ in the domain of $f$ if $$f'(a^-) \to +\infty$$ and $$f'(a^+) \to +\infty$$ Or both approach to $-\infty$. But for $f(x)=\sin^{-1}x$ $...
3
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2answers
55 views

For which values of $a,b$ is the matrix invertible?

I am trying to figure out the below question: 15. For which values of the constants $a$ and $b$ is the matrix $$A = \left[\begin{array}{cc} a & -b \\ b & a \end{array}\right]$$ ...
1
vote
0answers
21 views

Integral inversion

Say I know this function $$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$$ where $m(x)$ is a Fourier transform of an infinitely differentiable real function, whose maximal ...
0
votes
2answers
89 views

Can the cross product of two non-invertible matrices be invertible?

To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible? Just used to help understand a Linear Transformation assignment question, don't ...