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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Let $f = \frac{x}{x-1}$. What is ${f}^{-1}$? Show that f and ${f}^{-1}$ are symmetric about $y = x$.

Let $f = \frac{x}{x-1}$. What is ${f}^{-1}$? Show that f and ${f}^{-1}$ are symmetric about $y = x$. Finding the inverse of $f$ is easy enough. It actually turns out to be the same as $f$ itself. ...
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1answer
19 views

Inverses of Multivariable Functions from R^m to R^n

I was reading about Riemann Sphere and I found out that from the system: $$ u(x,y) = x/(x^2+y^2+1) \\ v(x,y) = y/(x^2+y^2+1) \\ w(x,y) = (x^2+y^2)/(x^2+y^2+1) \\$$ we can find inverses: $$ ...
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0answers
22 views

Inverse fourier transform.

I am currently learning fourier transforms on my own, and I am stuck on a current conceptual idea that I need help with. I know that the $\int e^{-2\pi ...
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1answer
28 views

Modulo multiplicative inverse of floating numbers

I have a floating value $k$ and an integer $P$ I want to calculate $(\dfrac{k}{\sqrt5}) \mod P$ How do I calculate it? PS: I know how to calculate MMI (Modulo Multiplicative Inverse of integer ...
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0answers
15 views

Inverted derivative question

I am wondering about the inverted function theorem $({\bf G}_{\bf F}({\bf x}))^{-1} = {1 \over {\bf G}_{\bf F}({\bf x})}$ If I have an equation ${\bf y} = \left({d {\bf F}({\bf x}) \over d {\bf ...
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1answer
35 views

Is a regular stochastic matrix definitely nonsingular?

Is a regular stochastic matrix definitely nonsingular (invertible)? How to prove it ? It says here that 'For a regular matrix always an inverse matrix exists' ...
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2answers
27 views

Inverse Trig Functions Composite functions of Csc, Sec, And Cot

Ok guys.. I'm trying to get prepared for my test tomorrow and I did numerous exercises. But I stumbled upon one of the "types" of exercises. Which is a composite function in $\csc$ and $\sec$. For ...
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0answers
28 views

Matrix Inverse Formula

Is there an alternative to David Lay's proof of the Matrix inverse formula for an n x n matrix? I find it difficult to grasp his steps; especially the interchanging of ith and jth subscript. Thanks
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0answers
16 views

Matlab function for finding matrix inverse with cayley hamilton theory

I want to write function in matlab that would calculate the inverse of a matrix using its trace. I know that there are other ways to calculate the inverse but I need it to be with trace. I couldn't ...
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0answers
28 views

Hessian Inverse

I have a problem to inverse Hessian matrix in MATLAB. What I did is I added eye(3)*1e-1 to the matrix elements, i.e., I have inv(Hessian1 + eye(3)*1e-1 ). This solved my problem and now my Hessian ...
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0answers
57 views

Find $\arctan(x) + \arctan(y)$ in terms of $\arctan((x + y)/(1 - xy))$ [duplicate]

Find $\arctan(x) + \arctan(y)$ in terms of $\arctan\left(\dfrac{x+y}{1-xy}\right)$. I want to essentially prove this equation given in the textbook: $$ \arctan(x) + \arctan(y) = \begin{cases} ...
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3answers
40 views

Homeomorphism & inverse, between $U=\{ (x,y) \in \mathbb{R^2} :|x|+|y|\leqslant 2 \}$ and $V=\{(x,y) \in \mathbb{R^2} : \max(|x|, |y|)\leqslant 3\}$

Find a homeomorphism, and its inverse, between $U$ and $V$ where: $U= \{ (x,y) \in \mathbb{R}^2 : |x|+|y| \leqslant 2 \}$ $V= \{(x,y) \in \mathbb{R}^2 : \max (|x|, |y|) \leqslant 3 \} $ I have ...
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1answer
35 views

Show that the special unitary group $SU(n)$ is a compact topological group

What I know: $SU(n)=${$A \in U(n): detA=1$} where $U(n)=${$n \times n$ matrices $A: AA^*=I=A^*A$} with elements in $\mathbb{C}$ and $A^*$ is the complex transpose of $A$ A topological group is a ...
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1answer
16 views

Analogue of right-inverse for non-surjective function

Given a function $f: X \to Y$, not necessarily surjective, is there a common name (and more concise definition than follows) for a function which maps elements in $Y$ where $f$ is defined to elements ...
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2answers
622 views

Example of a continuous function with a discontinuous inverse

What is an example of a function $f: \Bbb R^n \rightarrow \Bbb R^m$ such that $f$ is continuous and injective but that $f^{-1}$ is not continuous. Our professor teased us with the notion but I ...
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1answer
16 views

Having some trouble with inverse Laplace tranform

How to solve this using inverse Laplace transform? 1/[($s$+1)($s$+2)$^4$] I though of this solution which is $A$/($s$+1) + $B$/($s$+2) + $C$/($s$+2)$^2$ + $D$/($s$+2)$^3$ + $E$/($s$+2)$^4$ Then ...
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0answers
30 views

Inverse Laplace Transform of an Infinite Sum

How to find the Inverse Laplace Transform of the following expression $$1+\frac{-Xs^{2/a}-Ys^{3/b}}{1!}+\frac{(-Xs^{2/a}-Ys^{3/b})^2}{2!}+\cdots$$ Any approximation is also okay... Here $a$ and $b$ ...
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1answer
19 views

Differing graphs for simple inverse exponential problem

In class, we are learning exponential functions. The following inverse exponential problem is bothering me: $y=x^{-\frac{1}{9}}$. When graphed, I feel that it should look like it does on Desmos: ...
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0answers
28 views

Is my proof correct? (prequel to inverse matrices)

This question comes from a section before inverse matrices are introduced. Suppose $AD=I_m$. Show that for any b in $R^m$, the equation $A$x$=$b has a solution. [Hint: Think about the equation ...
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2answers
16 views

Inverse calculation

I am trying to project estimated internal resistance of a battery. We know that the internal resistance approximately halves as the capacity of the battery doubles. For example... A 2AmpHour cell ...
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2answers
32 views

Prove $\operatorname{arccosh}'(y) = \frac{1}{\sqrt{y^2 - 1}}$

I have to prove that Prove $\operatorname{arccosh}'(y) = \cfrac{1}{\sqrt{y^2 - 1}}$ for all $y \in (1, \infty)$ I have to do this using the formula for differentiation of inverse functions. ...
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2answers
35 views

What is the easiest way to find the inverse of the following block diagonal matrix?

Consider, for $\sigma^2_u, \sigma^2_e > 0$, the matrix $$\mathbf{X} = \begin{bmatrix} \sigma^2_u+\sigma^2_e & & & \\ & \sigma^2_u+\sigma^2_e\ \\ & & \sigma^2_u+\sigma^2_e ...
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1answer
35 views

Inverse Fourier transforms with Heaviside step function

I want to find the inverse Fourier transforms of: $$u(\nu + 1) \ \exp(-\nu)$$ Attempt: So the inverse Fourier transform is given by: $$\int^\infty_{-\infty} u(\nu + 1) \ e^{-\nu} e^{j2 \pi t} \ ...
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1answer
137 views

The integral of this horrible looking expression [closed]

Whats the $$\int (\cos(\tan^{-1}(\sin(\cot^{-1}x))))^2dx$$ no idea what to substitute already this is looking bad and that square is making things worse. Please help Thanks!!
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1answer
29 views

Inverse Fourier Transforms

Find the inverse Fourier transform of the following: $$\sin(2 \pi \nu T) \cos (10 \pi \nu T) / (\nu T)$$ Attempt: I was told it was easier if we rewrite this in terms of a $sinc$ function. I think ...
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1answer
29 views

Inverse of a piecewise function including max including max

Doing a hobby project of mine i have determined the following function. $a$ and $b$ are always known, and $p,a,b$ are all non-negative. $t(p) = p + \max(\max(p-a, 0) \cdot 1.05-b, 0)$ if $(p-a) \leq ...
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1answer
42 views

Ivertibility , positive definiteness of block tridiagonal matrix which arose from poisson 2-d discretization

I have the following matrix , \begin{equation*} \begin{bmatrix} T & -I & 0 & \cdots & \cdots & \cdots & 0 \\ -I & T & -I & \ddots & & ...
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0answers
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Is this geometric Interpretation of $Q^T$ being orthonormal if $Q$ is orthonormal valid?

I was reading the book - Linear Algebra and its Applications, when I saw - Remark 2. Since $Q^T = Q^{-1}$, we also have $QQ^T = I$. When Q comes before $Q^T$, multiplication takes the inner ...
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0answers
31 views

Find the inverted matrix of $A=(a_i,_j),a_{i,j}={\dbinom{j-1}{i-1}}$

Let be $A=(a_i,_j)\in M_{n+1}(\mathbb{R})$ defined for all $(i,j)\in [\![ 1,n+1 ]\!]^2$, by $a_{i,j}={\dbinom{j-1}{i-1}}$. Let's show this is invertible and determine its inverted matrix. To my ...
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1answer
44 views

Calculate $59x^{-1}\equiv 1 \pmod{63}$ [closed]

How can I calculate $$59x^{-1}\equiv 1\pmod{63}?$$ I only know that $59$ is prime.
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0answers
32 views

Asymptotic to $f^{-1}(f ' (x)) $?

Let $tr(n)$ be the triangular numbers and $te(n)$ be the tetrahedral numbers. $$g(x) := \sum \frac{x^n}{n! 2^{tr(n)}}$$ $g'(x) = g(\frac{x}{2}) $ Now consider the analogue $$ f(x) = \sum ...
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0answers
23 views

Sherman-Morrison formula and a sum of outer products

A specific form of the general Sherman-Morrison formula reads $(1+u v^T)^{-1}$ = $1 - \frac{u v^T}{1+v^T u}$ where $1$ is the identity matrix, $u,v$ are vectors (say with length n) and T denotes ...
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0answers
33 views

Classifying functions whose inverse do not have a closed form

My initial question contained about how to classify functions whose integrals and inverses do not have a closed form. But I found this question: How can you prove that a function has no closed form ...
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1answer
35 views

Proof of the Sherman-Morrison Formula

I was reading a few proofs for the Sherman-Morrison Formula, which states that if $A$ is invertible and $M = A + \mathbf{u}\mathbf{v}^T$, then $M^{-1}$ is given by: $$A^{-1} - A^{-1}\mathbf{u} ...
43
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2answers
4k views

Given $g(x)$ and $f(g(x))$, solve for $f(x)$.

I've hit a wall on the above question and was unable to find any online examples that also contain trig in $f(g(x))$. I'm sure I am missing something blatantly obvious but I can't quite get it. $$ ...
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1answer
25 views

Finding modular inverse (wrong approach)

I'm trying to find the modular inverse of $$30 \pmod{7} $$ I have tried using the Euclidean algorithm and it gave me the right answer, which is $x \equiv 6 \pmod{7} $. However, I tried using another ...
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0answers
53 views

Easy way to find an inverse in $Z_n$

Well, I'm solving for x in $11x=3$ in $Z_{12}$. And the way for me to do this is finding the inverse of 11 in $Z_{12}$. But to get the inverse, I've tried all possible elements in $Z_{12}$ so that ...
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2answers
36 views

Solving $3\times 3$ matrix equations:

I am familiar with finding the inverse of matrices, but struggle to formulate matrix equations. In this particular question, one is asked to find the elementary matrix E where $E*A = B$. $A$ is ...
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1answer
34 views

sum of matrix inverse problem

Recently, when I was reading matrix analysis, a formula confused me a lot: If $A+B$ is nonsingular, then the following is true, $$A(A + B)^{-1}B = B(A + B)^{-1}A$$ I tested some random ...
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1answer
34 views

Logarithmic to linear

Given this function: $$\frac{1.0}{1024.0} + \frac{x}{100.0} * \frac{1023.0}{1024.0} = y$$ $$10 * \frac{\log_{10}(y)}{\log_{10}(2)} = z$$ $$z * 100 = a$$ ...
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2answers
48 views

What kind of distribution in this chart?

Could you tell me what kind of distribution is this? Chart This is the data: ...
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1answer
53 views

Relationship between inverse of related matrices

Suppose I have a matrix $A \in \mathbb{R}^{m\times n}$ with $m \geq n$ and suppose that a matrix $G=(A^T A)^{-1}$ exists. Now suppose that I have an other matrix $B \in \mathbb{R}^{m\times m}$ that ...
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1answer
21 views

Is this system invertible?

$y(t) = \int\limits_{-\infty}^{\infty} \frac {x(t)^2}{x(t-1)} dt\\$ I was trying to prove or disprove the invertibility of this function. The only thing I could think of was differentiating it. But ...
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2answers
40 views

Compute the indicated power of a matrix

Compute the indicated power of the matrix: $A^8$ $ A = \begin{bmatrix}2&1&2\\2&1&2\\2&1&2\end{bmatrix} $ I calculated the eigenvalues: $ \lambda_1 = \lambda_2 = 0, \lambda_3 ...
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2answers
46 views

Is this kernel invertible ? $K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}$

Is the following Kernel invertible? $K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}, x\in[0,1],y\in [0, \infty)$ i.e. if $\int_0^1 K(x,y) f(x) dx=0 ,\forall y\in [0, \infty)$ can we conclude $f(x)=0,x\in ...
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0answers
31 views

Confusion regarding logic in paper, “A NOTE ON THE INVERSION OF POWER SERIES,” published in the AMS journal

I was reading "A NOTE ON THE INVERSION OF POWER SERIES" and was able to follow the paper's reasoning until the bottom of the second page, where it states: in fact we can calculate the power series ...
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1answer
19 views

Trace of Hermitian Positive Semidefinite Matrix

Well, the question I want to ask is as follows. Suppose A and B are Hermitian Positive Semidefinite (PSD) matrices, I wonder if it is possible to prove $Tr(A*(A+B)^{-1})\in (0,1]$ (if it is ...
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1answer
46 views

Finding inverse polynomial in $\mathbb{F}_p[x]/(\psi)$ with maple

I need help with maple. I want to invert the polynomials $(x^{361}-x)^2$ and $4(x^3+2x+1)^{19}$ with the help of maple. The problem is that we are working in $\mathbb{F}_{19}[X]$ and modulo ...
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0answers
44 views

'Stable' Ways To Invert A Matrix

So lets say that I need to invert a matrix that is generally dense and is poorly conditioned. What are some ways I can get an accurate inverse? Here are my candidates: SVD Inverse Inverse Via ...
0
votes
2answers
64 views

If a matrix $A^2$ is invertible, is $A^3$ invertible? [closed]

I know how to find out if a matrix $A^2$ is invertible if $A^3$ is invertible, but how can you find out invertibility if it's the the other way around?