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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64 views

Is there a way to simplify this inverse?

I have to compute an inverse of the form $$ (K + t(XO^T + OX^T))^{-1} $$ where $K,X,O$ are all $n\times n$, and $K$ is symmetric. Assume that $K$ is invertible and that $K^{-1}$ is known. I would ...
3
votes
1answer
67 views

Property of Matrix Inverse / Matrix Inverse Derivative

I am given real, symmetric matrices $X \succ B \succ A \succ 0$ (where '$\succ$' signifies positive definiteness such that if $B \succ A$ then $B-A \succ 0$ is positive definite). Further let the ...
2
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1answer
29 views

Inverse Matrices Where the Entries Are Variables

Is there a general formula, or a specific technique to find the inverse matrix of matrices where the entries are variables instead of numbers (is it even possible or defined)? For example, how does ...
1
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0answers
27 views

Bilateral inverse z transform

There is one problem that I am solving at the moment and I am a bit confused. I will explain what I do, and what I am thinking, so if there is mistake, please point them out. Problem : Find inverse z ...
1
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0answers
48 views

How can I invert/reverse a curve/ease function?

I have a range of values that represents a curve. This in turn is applied in programming to an interface - rotatable knobs to be precise. Let's say you have a knob that represents a value from 1-20. ...
0
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2answers
41 views

Show that for $\mathbb{N} = \{ 1, 2, 3, \ldots \}$, $s: \mathbb{N} \to \mathbb{N}$ where $s(n) = n + 1$ has infinitely many left inverses.

The exact textbook question is: Let $\mathbb{N}$ denote the set $\{1,2,3,\ldots,\}$ of natural numbers, and let $s:N \to N$ be the shift map, defined by $s(n) = n + 1$. Prove that $s$ has no right ...
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3answers
36 views

Consider a function $g: (-1, +1) \rightarrow \mathbb{R}$

Consider a function $g: (-1, +1) \rightarrow \mathbb{R}$ given by $g(x) = \frac{x}{1-|x|}$ Show that $g$ is 1-1 and find $g((-1,1))$ Find $g^{-1}$ Are $g$ and $g^-1$ continuous I found that $g$ ...
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1answer
36 views

Invertability of Non-Square Derivatives and Implicit Function Theorem

In the statement of the Implicit Function Theorem from Rudin, we are asked to assume as a condition that for some function: $$ f: \mathbb{R}^{m+n} \rightarrow \mathbb{R}^n$$ The derivative at some ...
1
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0answers
66 views

Invertability of a matrix

$\newcommand{\AA}{\mathbf{A}} \newcommand{\Tr}[1]{\operatorname{Tr}\left[#1\right]}$ I have a problem that I suspect there is a “relatively” simple answer to but it is currently eluding me. I am ...
0
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1answer
42 views

Remove the Kronecker operator in $\mathrm{trace((\Sigma^{-1}\otimes S^{-1})ZDZ^{T}})$

I am not sure if I can remove the Kronecker operator in the following formula $$\mathrm{trace((\Sigma^{-1}\otimes S^{-1})ZDZ^{T}}),$$ where $\Sigma,S, D$ are all positive-semidefinite and symmetric. ...
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3answers
142 views

Finding the inverse of a mapping that can be defined as a function on a specific domain

Let $A = \{x \in \mathbb R\mid x\geq2\}$ and $B = \{x \in \mathbb R\mid x\geq1\}$ and the function $f : A\rightarrow B$ is defined by $f(x) = x^2-4x+5$. With this domain and codomain, the function ...
1
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1answer
29 views

Inverse Laplace of $\frac {s}{RCs+1} $

I was wondering how you would be able to solve the inverse laplace of $$\mathcal{L}^{-1}\left\{\frac{s}{RCs+1}\right\}\left(s\right)\tag{1}$$ where $R$ and $C$ are constants?
0
votes
1answer
40 views

Showing bijective differentiable functions must have the same dimension in the domain and the image

$f:A\rightarrow B$ and $f^{-1}:B\rightarrow A, \quad A\subset \mathbb{R}^{\alpha} , B\subset \mathbb{R}^{\beta}$ are bijective differentiable, and I aim to show that $\alpha = \beta$. To do this i've ...
2
votes
1answer
63 views

Inverse of diagonally dominant matrix with equal off-diagonal entries

Is there an explicit expression for the inverse of strictly diagonally dominant matrix with identical off-diagonal elements? For example: $$ \begin{pmatrix} a & -b & -b \\ ...
1
vote
1answer
29 views

Invertible product of different-dimensional matrices

We have the following situation: $A$ is an $n\times m$ matrix, $B$ is an $m\times n$ matrix and $C$ is some invertible $n\times n$ matrix. Can we, in general, say $$A(BCA)^{-1}B=C^{-1}?$$ Clearly, if ...
0
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1answer
24 views

Inverse a simple equation

Consider equation $y = x\cdot 2^x$ Can you write $x$ based on $y$ ? Is it possible ? Thanks
2
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1answer
56 views

Proof that inverse of a matrix is unique [duplicate]

If B and C are both inverses of the matrix A,then B=C. Can't i prove it in following way ? Proof: AB=BA=I and AC=CA=I,then ...
2
votes
2answers
41 views

Necessary to prove the inverse is Invertible?

I am just starting out on linear algebra and I have come to a section the book that confuses me somewhat. The authour defines an invertible matrix A as: "A square matrix A is said to be invertible or ...
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2answers
23 views

An Inverse Laplace Transform Problem

I am having problems solving this inverse Laplace transform: ℒ$^{-1}\Large [\frac{s-3}{s[(s-3)^2+9]}]$ I did partial fraction decomposition, but ended up with complex expressions in some ...
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2answers
47 views

Invertible element of $S$

Let $S=\mathbb{Z}[\sqrt{2}]$ = {$a+b\sqrt2|a,b\in \mathbb{Z}$} and $R = \mathbb{Q}[\sqrt2]$ = {$\alpha + \beta\sqrt2 | \alpha, \beta \in \mathbb{Q}$}. Consider $x=3+2\sqrt2$ and $y = 3+4\sqrt2$ ...
0
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1answer
103 views

Properties of the inverse of unit (lower) triangular matrix

Is there any special properties about the inverse of a unit lower triangular matrix? I'm trying to prove this: $$L^{-1}=I_n + N + N^2 + ... + N^{n-1}$$ where $L$ is a unit lower triangular matrix ...
1
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2answers
44 views

Inverseof the function $f(x)=\frac{a^{2x}-1}{a^{2x}+1}$

I have problem to invert this function , Can anyone help me to solve it? $$f(x)=\frac{a^{2x}-1}{a^{2x}+1}$$ My attempt: change $x$ to $y$ and try to solve for $y$, but I could not.
0
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1answer
42 views

Inversing badly-conditioned square matrix: methodology

I have a badly-conditioned square matrix. I need to inverse it. For inversing, currently I'm doing the following steps: I take the badly-conditioned matrix with size of $n$ by $n$ By reduced row ...
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1answer
108 views

Finding the Modular Multiplicative Inverse of a large number

I am practicing some modular arithmetic and I am trying to find the multiplicative inverse of a large number. Here is the problem: 345^-1 mod 76408 I'm not sure how to go about solving this problem. ...
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1answer
34 views

inversing using Euclid's algorithm

The question is: Find the inverse of 14 mod 37. I don't know how to do, so could someone please explain it? Thanks in advance.
2
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0answers
91 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
0
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1answer
25 views

Linear algebra - projection matrix - inverse matrix

I am not sure how to prove this one: Let $A$ be a projection matrix so that $A^2=A$ and $A$ is not equal to zero. Find the inverse matrix of $I+cA$. Thanks.
2
votes
1answer
42 views

The inverse of the sum of two matrices in *Applied statistical decision theory *.

I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here. A theorem at the page linked states that if two matrices $A,B$ are non-singular and of ...
3
votes
3answers
206 views

“Orthogonal” Rectangular Matrix

Is it possible to have a matrix $\mathbf B \in \mathbb R^{m\times n}$ such that it satisfies: $$\mathbf B^T\cdot\mathbf B = \mathbf I_n$$ Where $\mathbf I_n$ is the $n\times n$ identity matrix. Or ...
5
votes
3answers
94 views

Determinant of the inverse matrix [duplicate]

I'm seeking for a proof of the following: Let $A$ be an invertible matrix. Then the determinant of $A^{-1}$ equals: $$\left|A^{-1}\right|=|A|^{-1} $$ I don't know where to begin the proof. Any ...
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1answer
59 views

Determinant of $\lambda I + A^TA$

What properties $\lambda I + A^TA$ have? I know that $A^T A$ is positive semi-definite, and symmetric. I want to show that the determinant of $\lambda I + A^TA$ decreases as $\lambda$ increases!
0
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4answers
38 views

Can we show that $K=\tan(\frac{\theta_B}{2} + 45^\circ)$, given $\theta_B = \arctan(K) - \arctan(\frac{1}{K})$?

I am studying two separate technical documents which are about the same topic. I would like to know if they are defining certain two variables exactly the same. In the first document, it defines a ...
4
votes
2answers
140 views

Invertibility theorem on the boundary for a function between two closed 2D manifolds

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected, closed domain $D\subset\mathbb{R}^2$ including its boundary $\partial D$. I am interested in the local invertibility of $f$ ...
0
votes
1answer
29 views

Proving that a matrix product is singular

I just played around in mathematica and found out that it seems like if $A$ is an $m\times n$ matrix and B is an $n\times m$ matrix, with $m>n$, then $AB$ is singular. How does one go about proving ...
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1answer
31 views

Inverse of a product in a group can be written as the product of the inverses of each element in reverse order

Let $(G,\circ)$ be a group and let $g_1,...,g_n\in G, n\in\aleph$. Prove that $(g_1\circ ...\circ g_n)^{-1}=g_n^{-1}\circ ...\circ g_1^{-1}$ I tried this by induction but was unsure how to take out ...
0
votes
1answer
22 views

Why is the CT system $y(t)=x(2t)$ invertible but its DT counterpart non-invertible?

Just for clarity, a system is invertible if distinct inputs lead to distinct outputs. That said, I have two systems, a continuous time system and a discrete time system: (1) $y(t) = x(2t)$ ...
0
votes
2answers
99 views

Can we find the inverse for a vector

Can we inverse a vector like we do with matrices, and why ? I didn't see in any linear algebra course such a concept of vector inverse and I was wondering if there is any such thing and if not, why.
2
votes
1answer
44 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
2
votes
1answer
35 views

How to find the inverse of the matrix over $\mathbb Z_5$

How to find the inverse of the matrix over $\mathbb Z_5$ $$ \left( \begin{matrix} 1 & 2& 0\\ 0 &2& 4 \\ 0& 0& 3\\ \end {matrix} \right) $$
0
votes
1answer
29 views

Next step to show that these matrice expressions are equal?

This is a problem from Discrete Mathematics and its Applications I know invertible means it is possible to take the inverse of this matrix. This is definition of a power of a square matrix from my ...
1
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1answer
33 views

Is the inverse of a causal function causal?

I am wondering if the inverse of a causal function is causal. I'll illustrate what I mean with an example: Assume $f$ is a bijection of $\mathbb R^2$ in $\mathbb R^2$. I assume $f$ is causal in the ...
0
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0answers
83 views

Inverse function theorem and Implicit function theorem.

I have been trying to prove that implicit function theorem implies the inverse function theorem. Be $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\det[DF(x_0)]\neq 0$ for $x_0 \in ...
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2answers
24 views

Order of Inverse Operations

so this is a very simple question but I am having a tough time with it. So it's finals week and I'm studying up for an Algebra 2 final. The only part I am having trouble with is finding the inverse ...
0
votes
3answers
146 views

Find all matrices where the matrix is its own inverse and the determinant is 1

I need to find all the matrices: $$\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ such that $$ad-bc=1$$ and $$A^{-1}=A$$ How would I go about doing this? I know that $AA=I^2$, ...
0
votes
1answer
12 views

Suppose that $p(x)=1/4x^4−2/3x^3-5/2x^2+6x-1/12 $withDom(p)=[1,2].Define$q(x)=p^−1(x)$. Show, algebraically, why q(x) exists

I don't know where to start. What does it means to define $q(x) = p^-1(x)$?
1
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1answer
47 views

Let $f(x) = \exp (x^2 − x + 6)$. Choose Dom(f) so that $f^{−1}$ exists. What is $f^{−1}$ and Dom($f^{−1}$) in your case?

I have already got $$y=\exp(x^2-x+16)$$ $$\ln y = x^2-x+6$$ $$\ln x=y^2-y+6$$ I know for getting inverse function we need to solve for $x$, but what should i do in this case?
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3answers
68 views

Invert a $2\times 2$ Matrix containing trig functions [duplicate]

Invert the $2\times 2$ matrix: \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} My thought was to append the $2\times 2$ identity matrix to the right ...
0
votes
1answer
41 views

Relation between $\tan^{-1}(x)$ and $\cot^{-1} (x)$

Suppose we've got $$I_1=\int_{-1}^{1} \tan^{-1}(x) + \tan^{-1} \left(\frac{1}{x}\right)$$ and $$ I_2=\int_{-1}^{1} \cot^{-1}(x) + \cot^{-1}\left(\frac{1}{x}\right)$$ So how can we relate $I_1$ and ...
0
votes
1answer
18 views

inversion of a symmetric matrix after that a column has been changed

Suppose $Z\in \mathbb R^{n\times k}$ and $S=Z^TZ$. Let now $Z(i, x)$ be the matrix $Z$ where the $i-th$ column has been replaced with $x$. Given $S^{-1}$ is there a quick way to compute ...
0
votes
1answer
30 views

Invertibility of $X^TX$ when sever multicollinearity in regression

I am studying about multicollinearity in regression and in the book it says, "if there is severe (but not perfect) multicollinearity, two or more predictor variables are highly correlated, so $X^TX$ ...