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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1answer
33 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 ...
1
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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 ...
7
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2answers
620 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 ...
1
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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 $b$ ...
0
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1answer
18 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 ...
0
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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. ...
2
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2answers
34 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 ...
0
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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} \ ...
0
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1answer
130 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!!
0
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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 ...
2
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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 & & ...
2
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0answers
28 views

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 ...
1
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0answers
30 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 ...
0
votes
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 ...
2
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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} ...
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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. $$ ...
1
vote
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 ...
0
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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 ...
1
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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 ...
1
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1answer
32 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 ...
0
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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: ...
1
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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 ...
2
votes
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 ...
0
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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 ...
0
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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 ...
1
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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 ...
2
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0answers
43 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
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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?
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2answers
50 views

Solving for $x$ in a Laplace equation

So I have this Laplace equation: $$s^{2}x+4sx+5=\frac{s}{s-1}$$ And I want to solve for $x$. My result is the following: $$x = \frac{5-4s}{s^{3}+3s^{2}-4s}$$ Which is also the same answer that for ...
0
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2answers
63 views

Finding all left inverses of a matrix

I have to find all left inverses of a matrix $A = \begin{bmatrix} 2&-1 \\ 5 & 3\\ -2& 1 \end{bmatrix}$ I created a matrix to the left of $A$, $\begin{bmatrix} a &b &c \\ ...
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3answers
64 views

Necessary and/or sufficient conditions for $A+B$ to be invertible

Let $A$ and $B$ be two $n\times n$ real invertible matrices. Are there necessary and/or sufficient conditions (involving only $A$ and $B$ separately, not $(A+B)$ iteself) for $A+B$ to be invertible? ...
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2answers
27 views

Inverse of the Cross Ratio for Mobius Transformation from Circle to Circle

I'm reading Conway's complex functions of one variable, and in chapter 3 he goes over Cross-Ratios. He defines the cross ratio to be $(z,z_1,z_2,z_3)=\frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}$, where ...
1
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2answers
37 views

What does it mean for f([x])=[2x] for a function mapping R/Z to R/Z?

Let X=R/Z (the circle), with a map $f : X → X$ given by $f([x]) = [2x]$. I'm a little lost on what $f([x]) = [2x]$ means. I thought the function was mapping the equivalence class [x] to the ...
1
vote
1answer
31 views

Linear Algebra - Real Matrix and Invertibility [closed]

Let $M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ be a real matrix $2n\times 2n$ with $A,B,C,D$ real matrices $n\times n$ that are commutative to each other. Show that $M$ is invertible if and only ...
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2answers
54 views

how to prove that invertible matrix and vectors span the same space?

Given $M$ is an invertible matrix, and {$\vec{v_1}...\vec{v_k}$} spans $R^n$, then {A$\vec{v_1}...A\vec{v_k}$} also spans $R^n$ What does matrix invertibility have to do with span?
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1answer
50 views

Inverse of linear combination of trigonometric functions [closed]

I have an equation of the form: $$\tan(y)=\alpha_1\cos(x)+\alpha_2\sin(x)$$ where $x$ and $y$ are in $(0,2\pi)$ and the coefficients are real numbers. Implicitly this defines $y$ as a function of ...
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0answers
40 views

Orthogonal matrix problem

So the question asks: Let $A$ and $B$ be n×nn×n orthogonal matrices, with $n≥2$. Which of the following matrices must be orthogonal? A. $A^TB$ B. The matrix C obtained by multiplying the second ...
0
votes
1answer
35 views

Show that Ax = b is solvable when [A b] is singular.

I have the following problem: Review: Suppose A is 5 by 4 with rank 4. Show that Ax = b has no solution when, the 5 by 5 matrix [A b] is invertible. Show that Ax = b is solvable when [A b] is ...
1
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2answers
35 views

Invertible function that “messes” order [closed]

I am looking for an invertible discrete function $f$ such that given some integer n, if i apply $f(i)$ for $i=0,\dots,n$ I would get all the integers in range $[0..n)$ exactly once, but in a "messy" - ...
0
votes
0answers
26 views

Inverse Functions (Discrete Math)

Say you have $f: \mathbb{Z} \to \mathbb{Z}$ defined by $f(x,y) = (2x+y, y)$ How would you check if the function was invertible? As well as determining it's inverse if it is? Thank you