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 ...

learn more… | top users | synonyms

0
votes
3answers
35 views

Functions - Inverses of graphs.

The question reads: sketch the graph of y=-3-x along with its inverse. From calculating the equation of the inverse graph, I come to y=-3-x, using the swap method. I then tried to plot both graphs ...
2
votes
1answer
31 views

Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
0
votes
1answer
14 views

for $1 \geq x \geq 0: {2x^2\over{(2+x)}} \leq y \Rightarrow x \leq \left(\frac{3}{2} y \right)^{1/2}$

So what I did is prove that $f(x) := {2x^2\over{2+x}}$ is increasing and then invert $f$ on $[0,\infty]$ this yields $(f\restriction_{[0,\infty[})^{-1}(y) = \frac{1}{4}(y+\sqrt{y}\sqrt{y+16})$ and ...
1
vote
0answers
24 views

Difficult examples of invertible, differentiable functions

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
2
votes
0answers
37 views

Inversion of a pairing function

I was reading this question on this site and I saw that the following pairing function was mentioned (a modified version of Cantor function): $$\langle x, y\rangle = x * y + ...
5
votes
1answer
139 views

Find Functions That Can Be Inverted from Their Sums

I have the following situation:$$ f_1(x_1) + f_1(x_2) + f_1(x_3) + \cdots + f_1(x_n) = c_1\\ f_2(x_1) + f_2(x_2) + f_2(x_3) + \cdots + f_2(x_n) = c_2\\ \vdots\\ f_n(x_1) + f_n(x_2) + f_n(x_3) + ...
4
votes
3answers
51 views

Inverse of an ordered pair?

Let $f: A \to B$ be a bijective function where $A = [0, 2\pi)$ and $B$ is the unit circle. Find the inverse of $f(\theta) = (\cos\theta, \sin\theta)$. I don't understand what it means to take the ...
0
votes
0answers
8 views

Linearization of hyperbolic function with unknown exponent

I have a graph that is clearly some inverse function of the form $y=Ax^n$ where n is a negative. I want to linearize this graph to give me the values of A and n without merely approximating the ...
-1
votes
1answer
36 views

Exponential Function Equation and inverse Pre-Cal

I am not completely sure if I wrote the equation correctly. For A I wrote: $m(t)=100(b^x)$ Not sure it is correct...but how do I find the inverse? That doesn't make sense to me. Do I use log?
5
votes
1answer
82 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
0
votes
1answer
18 views

Simplifying the inverse of the sum of 2 matrices

I would like to simplify the following inverse computation : $$(D+A)^{−1}$$ where $A=UΣU^T$ (eigenvalue decomposition). And D is a diagonal matrix such that $D = \lambda \boldsymbol{I}$ I know the ...
2
votes
1answer
50 views

Is there an easier way to find the inverse of a 3x3 matrix?

I know the normal process is to do row operations to transform the matrix to get the identity matrix and then apply the same row operations in the identity matrix to get the inverse. But this process ...
2
votes
1answer
36 views

Matrix inversion via Levi-Civita symbols

Using Cramer's formula for the inverse of a matrix $M_{ij}$, is it possible to express the entries $(M^{-1})_{ij}$ in terms of the entries $M_{ij}$ using the Levi-Civita symbol and Kronecker deltas? ...
6
votes
1answer
52 views

Show that $Y$ is invertible

Let X be a $40\times40$ matrix such that $X^3 = 2I$. I want to show that $Y= X^2 -2X + 2I$ is invertible as well. I tried working with the equations to see if I can get Y as a product of matrices ...
0
votes
1answer
23 views

inverse of sum of diagonal matrix and eigendecomposition

I would like to simplify the following inverse computation : $$(D + A)^{-1}$$ where $A=U\Sigma U^T$ (eigenvalue decomposition). And D is a diagonal matrix I know the inverse of A is ...
0
votes
1answer
91 views

Find the inverse of a piece-wise continuous function

I have the following piecewise continuous function: $$f(x)=\begin{cases}3x+1,~x\gt 0\\2-x^2,~x\leq 0\end{cases}$$ and I need to find its right inverse. Thus far I got that ...
1
vote
2answers
34 views

Why does the Gaussian-Jordan elimination works when finding the inverse matrix?

In order to find the inverse matrix $A^{-1}$, one can apply Gaussian-Jordan elimination to the augmented matrix $$(A \mid I)$$ to obtain $$(I \mid C),$$ where $C$ is indeed $A^{-1}$. However, I fail ...
1
vote
2answers
33 views

Prove that $\det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, (A+uv)^{-1}$.

Let $A$ be a $n \times n$ matrix, $u$ a $n \times 1$ matrix and $v$ a $1 \times n$ matrix. If $A$ and $(A+uv)$ are invertible, prove that $$ \det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, ...
0
votes
1answer
23 views

Searching for a function where the inverse exists in a neighborhood of a point, but the Jacobian is zero.

I'm looking for a function f in $\mathbb{R}^2$ such that the inverse function therem at some point P = (x,y) does not give an answer of whether the function is invertable in some neighborhood of P, ...
0
votes
0answers
20 views

Problems with the inverse of a banded matrix: not invertible?

I am creating with a software a banded matrix, which is also symmetric. In fact, its definition comes from an array, Array[q], whose length is ...
1
vote
1answer
67 views

Find the multiplicative inverse of $5$ in $\mathbb Z_{73}$

I'm having some trouble with this question. The inverse should result in $44$ but I am getting $29$ $$73 = 14 \times 5 + 3$$ $$5 = 1 \times 3 + 2$$ $$3 = 1 \times 2 + 1$$ so $\gcd(73,5)=1$ using ...
0
votes
0answers
21 views

Invert an Excel function containing the tangent

In the following excel formula: =95*1*1/TAN(RADIANS(M3-(10.3/2.01)))/5280) $M3=2.63715$ and let's say the result of this formula is: $5.508306483$ What would ...
8
votes
1answer
86 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
1
vote
1answer
57 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
2
votes
2answers
45 views

Use Euclid's algorithm to find the multiplicative inverse $11$ modulo $59$

I was wondering if this answer would be correct the multiplicative of $11$ modulo $59$ would be $5$ hence $5\cdot11 \equiv 4 \pmod{59}$. Is this correct?
3
votes
3answers
45 views

What does inversion mean?

I am in highschool taking some advanced math courses and I have some questions about terminology. There appears to be more definitions to the meaning of inversion in math than I can count. I'm ...
0
votes
0answers
35 views

Inverse Matrix in 3D

Suppose we have the following matrix in three dimensions $$ M_{ij} = g_{ij} + e_{ijk}z^{k} $$ where $e_{ijk}$ is an antisymmetric density, i.e. $e_{ijk} = \sqrt{\det g}\cdot\epsilon_{ijk}$ and $z^{k}$ ...
-1
votes
1answer
48 views

Questions on the formula for 2x2 inverse

Hi I was working on inverting 2by2 matrix in general form by using a,b,c,d. I know the formula (which is below) but I have questions in the process of getting the formula. 1) To get rid of the ...
0
votes
1answer
54 views

If two invertible matrices A and B commute, then A^-1 and B^-1 must commute as well ??

If two invertible matrices A and B commute, so their inverse must commute as well or not ?
0
votes
1answer
39 views

How to calculate inverse of Variance Gamma call price formula using Newton-Raphson search

The Variance Gamma call price formula is given by: $$C(0)= \int\gamma(R) e^{-rT} \int f\left(S(0) e^{\theta R+\omega T+\frac12 \sigma^2 R} e^{rT-\frac12 \sigma^2 R+\sqrt{T}\sqrt{R/T} \sigma ...
0
votes
1answer
24 views

Derivative of inverse cosecant?

I am slightly confused by this, because when I worked out the derivative of arccosec(x), my answer was $\frac{-1}{x\sqrt{x^2-1}}$, which agrees with the answers online. However this would imply that ...
4
votes
1answer
293 views

Inverse of the sum of the inverse of two matrices

I need to compute $ (A^{-1} + B^{-1})^{-1} $. Both $A$ and $B$ are symmetric and $A$ is invertible and PSD. I already know $B^{-1}$ and $A$, but I don't have $A^{-1}$ and $B$. Is there a formula to ...
0
votes
1answer
54 views

Relation between two inverses

Suppose you know $(I+T)^{-1}$, is there any way for approximate the inverse of the matrix $(I+\alpha T)^{-1}$, where $\alpha\in{\mathbb{R}}$?
1
vote
2answers
14 views

Inverse of matrix with 1 in diagonal and some entries above them.

Suppose matrix N has a,b,c above the main diagonal, and all other entries equal to $0$. that is, $N=\begin{bmatrix} 0 & a & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & 0 & 0 & c ...
1
vote
2answers
34 views

Evaluate cos[(1/2)[arcsin(-3/5)]]. I'm not sure what i'm doing wrong.

$x=\arcsin(-3/5), \; \sin x = -3/5$ **Drew a triangle to find $\cos x$ $\cos x = 4/5$ Now, I don't know what to do from here. I know I have to use a double angle formula, but when I evaluate the ...
0
votes
0answers
23 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 ...
1
vote
0answers
51 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 ...
-1
votes
1answer
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 ...
2
votes
0answers
28 views

How to solve an inverse relationship (cooking temp/time)

How to figure out exactly the "add a little more time" to the question: cook at 425 deg for 18 minutes ... if I have several things in the same oven and need to set the oven at 375. I can't use a ...
2
votes
2answers
51 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 ...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
1answer
47 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. ...
1
vote
1answer
31 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 ...
1
vote
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 ...
0
votes
2answers
28 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 ...
0
votes
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 ...
1
vote
1answer
53 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 ...
1
vote
2answers
26 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 ...
2
votes
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)| ...