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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Finding the domain of a difficult inverse

$f(x)=\frac{3x+5}{-6x+2}$ , largest possible domain Find $f^{-1}(x)$ of this 1-1 function and the domain. So I wrote the equation as $$y=\frac{3x+5}{-6x+2}$$ Interchanged x and y, and made y ...
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40 views

Is $T(M)=PMP^{-1}$, where $P=\begin{bmatrix}2&3\\5&7\end{bmatrix}$ linear? If so, how to prove?

If I define $\vec{v}=\begin{bmatrix}a\\b\end{bmatrix}\text{and }\vec{w}=\begin{bmatrix}c\\d\end{bmatrix}$, I end up getting ...
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44 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...
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20 views

Row sums of inverse of PageRank matrix variant

In the book "Deeper inside Pagerank" (Amy N. Langville and Carl D. Meyer), (http://www.ulco.nl/docs/Langville.pdf), in the page 352 of the book (page 18 of the document in url), it is stated that "The ...
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38 views

Application of Inverse Function Theorem

This is a seemingly easy exercise. Yet I am not sure if I am missing any finer details here as this is listed as one of the challenging problems on Dr. Epstein's (Upenn) course site for real analysis. ...
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1answer
35 views

$y=N(N^TT N)^{-1}N^TT$

Let $T$ be a square $n\times n$ matrix. This matrix is symmetric and positive definite. Let $N$ be a $n\times s$ matrix where $s<n$. I want to be able to compute: $$y=N(N^TT N)^{-1}N^TT$$ I can ...
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29 views

How to calculate frequency with clock signal is 500ps in digital logic?

How can i calculate frequency if clock signal 500ps. I know the only formula, that is T=1/f But i cant able to calculate, can ...
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38 views

find the inverse of $\frac{1-e^t}{1+e^t}$

Hi I am trying to prove that the inverse of $f(t) = \frac{1-e^t}{1+e^t}$ is $F^{-1}(t) = \ln\left(\frac{1-t}{1+t} \right )$ But I don't quite know where to start? Do I just sub ...
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1answer
70 views

find an inverse function of complicated one

Let $f:\mathbb{R}\rightarrow \mathbb{R}$: $$f(x) = \sin (\sin (x)) +2x$$ How to calculate the inverse of this function? So far i searched a lot in the internet but i didn't find any easy algorithm ...
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137 views

Understanding inverse of a function

I was trying to understand the proof for the following proposition. Proposition: If $\{f_n\}$ is a sequence of $\bar{\mathbb{R}}$ valued measurable functions on $(X,\mathcal{M})$, then the functions ...
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3answers
73 views

Proof Regarding Determinants of a Matrix

Prove the following statement: If $A$ is an $n$ by $n$ matrix, such that $\sum_{j = 1}^n a_{ij} = 0$, for all $1 ≤ i ≤ n$, then $\det A = 0$ too. (Sorry I don't know how to format this equation) ...
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39 views

Invertible matrix problem

Given three $n \times n$ matrices $A$, $B$ and $C$. Prove that if $AB+AC$ is an invertible matrix then $A$ is also an invertible matrix. How can this be possible? I found that $B=A^{-1}-C$ and when I ...
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0answers
31 views

Sherman–Morrison–Woodbury formula and hollow matrix

Suppose there are two matrices: $A_{n\times n}= \begin{bmatrix} a_0 & 0 &a_1 & \dots \\ 0 & a_1 & 0 &\dots \\ a_1 & 0 &a_2 & \dots \\ \vdots & \vdots & ...
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1answer
37 views

Inverse Laplace tranform via the table formulas

In my inverse Laplace table there is this inversion "formula": $(1) \frac{1}{s-a} \rightarrow e^{at}$ I understand that $\mathcal{L}^{-1}[\frac{1}{s+4}] = \frac{1}{2}\sin(2t)$ But why can I not do ...
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27 views

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x?

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x? Solution $\tan^{-1}x=\tan^{-1}a-\tan^{-1}b=\tan^{-1}\frac{a-b}{1+ab}$ ...
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1answer
55 views

Tan inverse summation

$$S=\sum\limits_{i=1}^{4}\tan^{-1} x_i$$ How to simplify this ? I think I will have to use this : but it looks too long a method . Is there a method or symmetrical way which yields ...
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1answer
42 views

Arithmetic modulo primes task

I'm dealing with a problem here. The problem is as follows: There is a set $Z_p=\{0,1,2,3,...,p-1\}$ where $p$ is a prime. From this set we form a new set $B=\{x+x^{-1}\mid x\in Z_p\}$, where the ...
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82 views

Closed form for elements of inverse matrix of lower triangular matrix of any size

If we have a lower triangular matrix $$A=\left(\begin{array}{rrrrr}a_{1,1}&0&0&\cdots&0\\a_{2,1}&a_{2,2}&0&\cdots&0\\a_{3,1} &1_{3,2}&a_{3,3}&\cdots&0\\ ...
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20 views

a matrix inverse problem

Given a matrix $X$, let $D$ be a diagonal matrix whose diagonal elements are row sums of $X$, let $I$ be an identity matrix. Now I have a resultant matrix of $Y=(I-X)^{-1}$, and I would like to ...
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24 views

Deriving an identity using the Woodbury matrix identity

I am working through an algorithm derivation in Kernel Adaptive Filtering: A Comprehensive Introduction by Liu, Principe and Haykin. The part I'm having trouble with is on page 104 if you have the ...
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13 views

Inversion of Boolean function Application

Asume you have a boolean function $f$ which takes $n$ parameters and gives $m$ results. In addition, you have a boolean function $g$ takes $p$ parameters and gives out $n$ results. You could ...
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48 views

Show that the inverse function to $f(x)=\int_{1}^{x}\frac{dt}{t}$ is differentiable

Show that the inverse function to $$f(x)=\int_{1}^{x}\frac{dt}{t}$$ is differentiable. I know that the integral is $\ln(x)$, but I don't know how to show that it is differentiable in a good way ...
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53 views

Cholesky, Inverse, and Determinant when updating the diagonal of a symmetric positive definite matrix

Suppose that $A$ is a symmetric positive definite matrix and assume its dimension $n$ is large. Let $I$ be the $n \times n$ identity matrix and $m \neq 0$ be a scalar. I'm interested in computing as ...
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56 views

Word problem about finding the inverse derivative

I have the following word problem. I need to find and interpret the meaning of the inverse derivative of a function. At a gas station, the function f(p) is the number of gallons of gasoline sold when ...
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4answers
64 views

Calculate inverse of matrix

If $$A=\begin{bmatrix} -5 & 1 & 0 & 0\\ -19 & 4 & 0 & 0\\ 0 & 0 & 1 & 2\\ 0 & 0 & 3 & 5\\ \end{bmatrix}, $$ how do I calculate $A^{-1}$? Is there any ...
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26 views

Is it Possible to Develop an inverse function using the function it self

Is it Possible to Develop (taylor expansion) of an inverse function by knowing the function it self ? If Yes ,Can you illustrate with a simple function I know that we use the identity formula $$ ...
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44 views

Finding a Matrix from Determinants

I've stumbled upon this problem on my homework, and I have no clue how to do it, and haven't found any help online: If I'm understanding this correctly, then $det(M) = ad - cb + eh - gf$ ? What I ...
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146 views

Calculate inverse of arbitrarily sized, lower triangle matrix with a specific pattern.

I have a matrix of the following form: $$A=\begin{bmatrix} 2 & 0 & 0 & 0 \\-1 & 2 & 0 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & -1 & 2 \end{bmatrix}$$ which, in ...
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45 views

Inverse of a special matrix

Is there easy (analytical) way to find the inverse of the following matrix, where $C$ is a vector? $$ \begin{bmatrix} 1 & C^\top \\ C & CC^\top \end{bmatrix} $$
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22 views

Derivative of inverse function where inverse is known only numerically.

I have the following polynomial function in $\mathbb{R}$: $$f(x,a)=ax^3+x$$ However, $x$ also depends on $a$, so we should rather write: $$f(x,a)=ax(a)^3+x(a)$$ Now I need derivative of $f$ and ...
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36 views

Inverse of split functions

I don't know how to find the inverse is of a function when is split. Example, $\Bbb R_+$ is the set of positive real numbers. $f : \Bbb R \to \Bbb R_+$ $$f(x) = \begin{cases} 2-x & \text{if } ...
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25 views

check a matrix for positive semidefiniteness in the general case?

Here is the problem that I am facing. I know that for a positive semidefinite matrix to exist, the condition below must be satisfied: $x^\intercal Y x \geq 0$ where $x$ is a non-zero vector, i.e. a ...
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1answer
47 views

What does it mean for a matrix to change basis?

I understand what it means for vectors, i.e. $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} ...
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57 views

How to calculate the inverse of sum of a Kronecker product and a diagonal matrix

I want to calculate the inverse of a matrix of the form $S = (A\otimes B+C)$, where $A$ and $B$ are symetric and invertible, $C$ is a diagonal matrix with positive elements. Basically if the ...
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31 views

jacobian matrix of $f^{-1}$ at $(1,0,2)$ given $f(x,y,z)$

If $f(x,y,z)=(sin(xyz),(x+x^2)*cos(y),y)$ and $f$ has a local inverse in the neighborhood of $(0,1,1)$, how do I find the jacobian matrix of this inverse at $(1,0,2)$? I know from definition that ...
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57 views

Find the slope of the tangent line to the graph of $f^{-1}$

Given function $f$, find the slope of the line tangent to the graph $f^{-1}$ at the point on the graph $f^{-1}$. $f(x)=\sqrt{5x}$; $(4,\frac{16}{5})$? Here is what I have thus far: $f'(x)= ...
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84 views

How to solve an Inverse differentiation problem

If f is a one-to-one function where $f(3)=2$ and $f'(3)=6$, what is the value of $(f^{-1})'(2)$? I am not even sure where to start with this question. I was hoping someone can help $f$ of $3 =2$ and ...
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28 views

implementing modular multiplicative inverse.

I wish to implement the clifford cocks algorithm using GMP. In the encryption part: $c_1=t_1+at_1^{-1}\bmod n$. Following $( a b \bmod n ) = ((a \bmod n) \cdot (b \bmod n ))\bmod n$, I took the ...
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35 views

Integral of reciprocal of a piecewise linear function

Let, e.g. $$ f(x) = \begin{cases} x,\quad x<1, \\ 1,\quad x\geq1, \end{cases} $$ a piecewise linear function. Does the following hold for $g(x) = 1/x$? $$ \begin{align} g(f(x)) ...
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35 views

what function fulfills these conditions? [duplicate]

So I know that if $f(x) = x^{-1}$, than $f(f(x)) = x$ but $f(x)$ is not necessarily $x$. So now, is there $g(x)$ such that $g(g(x)) \neq g(x) \neq x$ but $g(g(g(x))) = x$? If so what is it, else why ...
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59 views

Inverse of 3-by-3 matrix

Hi, so this question is taken straight from khan academy help exercises, i know how to do it dynamically meaning using the determinant and the adjugate how i was trying to do it using guass bla bla ...
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prove that $X$ is invertible if and only if $Y$ is invertible. if $(-1)^i(1+i)x_i^T=y_i$

$X=[x_1,x_2,...,x_n]$ and $Y =$ $y_1\\y_2\\...\\...\\...\\y_n$ where $x_i$ and $y_i$ are column and row matrices respectively. $X$ and $Y$ are both $n$ x $n$ matrices. if $$(-1)^i(1+i)x_i^T=y_i$$ ...
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1answer
51 views

Find fundamental matrix of a 2x2 matrix with rank 1

$$ x'(t) = \left[\begin{array}{cccc}0&1\\0&t\end{array}\right]x(t)$$ I am having trouble computing the fundamental matrix. I get: $$ x1(t) = x2(0)*exp(0.5t^2) $$ $$ x2(t) = x2(0)*exp(0.5t^2) ...
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33 views

Inverse Sine and cosine

$\arcsin(\cos(x))=1/2$ Find $x$. I got $-1/2$ or $2\pi-1/2$, but I don't know the correct answer. I tried graphing unit circle.
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58 views

A square matrix with the diagonal and antidiagonal elements different from zero. Looking for some already known property.

I am interested in the properties of a matrix with elements different from zero only on the main diagonal and antidiagonal, like this: $$ \begin{matrix} a & 0 & 0 & h \\ ...
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1answer
59 views

If we have a square matrix thats invertible, do its row and column space coincide?

If we have a square matrix thats invertible, do its row and column space coincide? Regarding an nxn invertible matrix: -The row space of the matrix is R^n -The column space of the matrix is R^n ...
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32 views

compute the inverse function

Assume $h(x)$ is an invertible function. Let $g(x)=2+8h(4x+1)$. Find the inverse of $g$ in terms of $h^{-1}$ So following the usual steps to get the inverse function, I rearranged to get ...
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66 views

Show a matrix is invertible [duplicate]

How to show that $$A=\begin{pmatrix}1233&2344&1324&3456\\ 2342&11233&1432&13256\\234132&32432&1234567&43254\\423412&42354&452356&13245\end{pmatrix}$$ ...
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4answers
80 views

Calculate the multiplicative inverse modulo a composite number

I want to calculate $ 8^{-1} \bmod 77 $ I can deduce $ 8^{-1} \bmod 77$ to $ 8^{59} \bmod 77 $ using Euler's Theorem. But how to move further now. Should i calculate $ 8^{59} $ and then divide ...
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70 views

Summation of $\tan^{-1}$ series

I am given $$S=\sum\limits_{n=1}^{23}\cot^{-1}\left(1+ \sum\limits_{k=1}^n 2k\right)$$ On expanding the sigma series becomes $$S= 23\cot^{-1}(3)+22\cot^{-1}(5) + \cdots + \cot^{-1}(47)$$ And in tan ...