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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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 ...
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55 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 \\ ...
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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 ...
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24 views

Inverse a simple equation

Consider equation $y = x\cdot 2^x$ Can you write $x$ based on $y$ ? Is it possible ? Thanks
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52 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 ...
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39 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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22 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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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 $y = 3+4\sqrt2$ ...
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76 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 ...
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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.
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41 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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79 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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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.
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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 ...
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1answer
24 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.
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1answer
38 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 ...
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3answers
174 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 ...
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88 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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50 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!
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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 ...
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139 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 $f$ ...
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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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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 ...
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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) $y(t) = x(2t)$ ...
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89 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.
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42 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} ...
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35 views

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

How to find the inverse of the matrix over $\mathbb Z_5$ $$ \left( \begin{matrix} 1 & 2& 0\\ 0 &2& 4 \\ 0& 0& 3\\ \end {matrix} \right) $$
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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 ...
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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 ...
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68 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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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 ...
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126 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 $AA=I^2$, ...
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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 $q(x) = p^-1(x)$?
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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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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 ...
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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 ...
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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 ...
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28 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 $X^TX$ ...
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Existence of continuous angle function $\theta:S^1\to\mathbb{R}$

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all ...
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65 views

Given A is a nil-potent matrix (given $ A^k=0 $), prove that A-I is invertible. Is my proof correct?

Given $A$ is a nil-potent matrix (given $A^k=0$), prove that A-I is invertible. I have proved the statement using contradiction, and I want to know if it is correct: Let $ A-I \neq I.$ ...
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Inverse of a unipotent matrix

Show that all unipotent matrices are invertible. Also, specify a formula for the inverse of a unipotent matrix. Now, I've tried to approach the problem using the determinant: a matrix is unipotent, ...
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Invertible “Sigmoid + x” function

I need an invertible function that represents a smooth transition between two straight, parallel line segments, like this: Depicted is $f(x) = -0.3/(1+e^{-10*(x-p)})+0.3/2+x$ (where $p$ is the ...
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33 views

Would there be no input or input does not exist?

This problem is from Discrete Mathematics and Its Applications. And the definition of inverse from the book: For part 43 (c), would the inverse not exist? For the floor function, in terms of $f(a) ...
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inverse of a power series with one specific solution

I have a school assignment and for now, I don't know where to start, I have to show that there exist a surrounding $U$ of $0$ where the following is true: If $y\in U$ , the equation $y=\frac{x}{f(x)}$ ...
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44 views

Matrix Inversion acceptable Condition Numbers

When considering matrix inversion it is worth while worrying about the condition number of the matrix you wish to invert. Matrices that are poorly conditioned can often create inaccurate results. This ...
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An element $a$ of a monoid $M$ is invertible iff there exists $x\in M$ such that $axa=1$

An element $a$ of a monoid $M$ is invertible iff there exists $x\in M$ such that $axa=1$ I can't do this one. How do I get started? It looks like it is saying there is only an inverse if ...
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84 views

Finding inverse of $f(x) =\frac{\ln(x)}{x}$

How do you find the inverse of the following function $$f(x) = \frac{\ln (x)}{x}$$ What looked like a simple question made my head hurt during exam.
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Why is the left inverse of a matrix equal to the right inverse? [duplicate]

Given a square matrix $A$ that has full row rank we know that the matrix is invertible. So there is a matrix $B$ such that $$ AB=1 $$ writing this in component notation, $$ A_{ij}B_{jk}=\delta_{ik} ...
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Given a finite metric space, are the matrices of triangle inequality errors invertible?

I have been working on some problems regarding finite metric spaces and have already proven/positively answered the following statement/question if the underlying metric has additional properties. Now ...
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40 views

complex and decimal tetration

So recently in the blog post on tetration, it talked about tetration with nice clean powers (calling them these because I don't know the right term). But how does it work when given a complex power? ...