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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24 views

Inverse function for a surface of revolution

I have the following function: $$ f(x)= c_1\cdot c_2\cdot x\cdot \arctan\left(c_2\cdot x\right)-\frac{1}{2}\cdot c_1\cdot \ln\left(1+c_2^2\cdot x^2\right) $$ with $c_1=0.003$ and $c_2=150$ constants ...
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53 views

a special matrix inverse

Let $A=\left( \begin{matrix} {{A}_{11}} & \ldots & {{A}_{1n}} \\ \vdots & \ddots & \vdots \\ {{A}_{n1}} & \cdots & {{A}_{nn}} \\ \end{matrix} \right)$ be an ...
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23 views

Inverse of function with two Exponential Eulers Terms

How can I go about getting the inverse of$ f(t) = e^{-.001t}\cdot e^{-.005t}$? I have found a couple of calculators online that suggest that the answer is: $t=-166.667\ln(y)$, but I would like to know ...
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2answers
55 views

Solving $z=w/2-\sin(tw)/(2t)$ for $w$

Is it possible to solve $$z=\frac{w}{2}-\frac{\sin(tw)}{2t},$$ for $w$? My first thoughts were that we would have to be careful about the domain of $f(w)$ so that the inverse was actually a function ...
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1answer
36 views

help with inverse function in $\mathbb R^2$

$F(x,y)=(x^2+2y^2,2x^2+y^2)$, and $A=\{(x,y):x>0,y>0\}$ I need to show $F(A)=\{(u,v):0<u/2<v<2u\}$ I also need to find what is $G(=F^{-1}):B\rightarrow A$ For the first question I ...
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19 views

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$ I need to find the following: $(a)$ Show $F$ is one-to-one on $A$. $(b)$ Show that $F(A) = \{(u,v) : 0 < \frac{u}{2} < v ...
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1answer
62 views

Matrix derivatives of determinant and inverse related to $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$

I would like to calculate the derivatives of determinant and inverse related to the term $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$ with respect to $\mathbf{X}$, where $\mathbf{C}$ is a constant matrix. ...
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1answer
41 views

Inverse symmetric circulant matrix

I want to inverse a very particular matrix numerically. The matrix is always symmetric and circulant. As an example of a 4x4 matrix I would want to inverse \begin{pmatrix} v_0 & v_1 & v_2 ...
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1answer
34 views

Inverse Function Theorem when determinant is undefined

For $f(x,y) = (x^3 - y^2, \sin{x} - \ln{y})$ f-inverse exists and is differentiable in a non-empty set around $(-1,0)$. Find $D(f^{-1})$ at $(-1,0)$. Seemingly this is an Inverse Function Theorem ...
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2answers
32 views

Inverse Image Proof

Let $f:X\rightarrow Y$. Let $A$, $A_1$ and $A_2$ be subsets of $X$ and $B$, $B_1$, and $B_2$ be subsets of $Y$. Then, I need to prove that $f^{-1}(B_1\cup B_2)=f^{-1}(B_1)\cup f^{-1}(B_2)$. I know ...
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2answers
34 views

What is needed to apply the inverse function theorem to $f(x,y,z) = \left(\frac{ax^2 + by^2}{2}, \frac{cy^2+dz^2}{2}, \frac{ex^2 + fz^2}{2} \right)$?

Let $f:\mathbb{R}^3 \to \mathbb{R}^3$ be $$f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right).$$ My question is what restrictions are necessary on ...
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1answer
60 views

Inverse of matrix mod $26$ wolframalpha wrong

I want to find $A^{-1} \pmod{26}$ for $A=\begin{bmatrix}10&3\\5&3\end{bmatrix}$ and I did the conventional $\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ and found the ...
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2answers
29 views

Is there any way to test the existence of left or right inverse matrix?

I know that the inverse matrix of a square matrix exists iff its determinant isn't 0. What about a non-square matrix? Is there any theorem about the existence of a ...
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1answer
79 views

Prove $\frac1{\sqrt x}$ is continous on $(0,\infty)$. Stuck on last line!

Let $f(x) = \frac1{\sqrt x}$ for $x\in(0,\infty)$. Given $\varepsilon>0$ and $x_0\in(0,\infty)$, show there exists $\delta>0$ such that $$|x-x_0|<\delta$$ implies that $$|f(x)-f(x_0)| ...
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1answer
36 views

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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1answer
41 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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2answers
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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22 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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1answer
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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1answer
38 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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1answer
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
72 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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2answers
142 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
76 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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42 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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34 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
38 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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1answer
28 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
59 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
44 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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85 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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1answer
21 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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1answer
33 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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1answer
14 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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1answer
76 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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1answer
67 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
69 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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1answer
27 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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45 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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3answers
159 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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2answers
46 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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27 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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1answer
42 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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29 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
50 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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2answers
71 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 ...
0
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1answer
32 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 ...
2
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3answers
60 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)= ...