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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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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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
141 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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3answers
66 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
28 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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17 views

How to prove that a matrix with specific property is invertible?

If we have a square matrix $$ M = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & ...
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Invertibility condition for $f:\mathbb{R}^2\to\mathbb{R}^2$ on the domain boundary

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected domain $D\subset\mathbb{R}^2$ with a smooth boundary $\partial D$. I am interested in the local invertibility of $f$ in a ...
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21 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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24 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 ...
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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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2answers
74 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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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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1answer
30 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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20 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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1answer
28 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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21 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
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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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3answers
85 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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43 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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38 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 ...
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15 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 ...
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1answer
13 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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126 views

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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55 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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54 views

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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30 views

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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1answer
29 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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1answer
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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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27 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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67 views

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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1answer
80 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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5answers
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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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22 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? ...
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1answer
34 views

Proving this function is an open map

Prove the function $f(x, y, z) = (x^3, y^2-z^2, yz)$ is an open map from $\mathbb{R^3}$ to $\mathbb{R^3}$ (i.e for every open set $U$ of $\mathbb{R^3}$, $f(U)$ is open). I know, as an application of ...
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1answer
32 views

Solving equations with matrices

Say I have $4$ simultaneous equations \begin{cases} 4.3S_1 - P = T \\ 8S_2 - P = T \\ 5.5S_3 - P = T \\ S_1 + S_2 + S_3 = T. \end{cases} I'm trying to solve these in Excel using MINVERSE and MMULT ...
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If $f(AB) =f(A)f(B)$, then $A$ is inversible iff $f(A)\neq 0$

Let $f:\mathscr M_n(\mathbb K) \to \mathbb K$ be a non constant function such as $f(AB) = f(A)f(B)$ for all $A,B$ in $\mathscr M_n(\mathbb K)$. The question is to show that $M\in GL_n(\mathbb K)$ iff ...
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Inverse transversal is perfect if and only if $li=i^0lil^0$ for $i \in{I}$, $l \in{\Lambda}$

I am attempting show that an inverse transversal is perfect if and only if $li=i^0lil^0$ for $i \in{I}$, $l \in{\Lambda}$. An inverse transversal is perfect if $li(li)^0=(li)^0li$. I have shown the ...
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187 views

find x where $x^{11} \mod 41 = 10$

In a previous part of the question, I am asked to find $11^{-1} \mod 40$. I've done that, the answer's $11$. The question continues: find $x$ where $x^{11} \mod 41 = 10$ showing how you could get ...
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3answers
55 views

Solve logarithmic equation for $x$ to find the inverse of $f(x)= \ln(x+\sqrt{x^2+1})$

Let $f(x)= \ln(x+\sqrt{x^2+1})$. Find $f^{-1}(x)$. Here is what I got so far: $y= \ln(x+\sqrt{x^2+1})$, rewrite as $x= \ln(y+\sqrt{y^2+1})$, then $$e^x= y+\sqrt{y^2+1}$$ $$e^x-y= \sqrt{y^2+1}$$ ...
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1answer
73 views

Questions about matrix rank, trace, and invertibility,

(a) Prove that a square matrix $T$ of rank one has $\text{tr}(T)=0$ if and only if $T^2=0$. (b) Consider a matrix $A$ of the form $A=aI+T$, where $a\ne0$, $I$ is the identity matrix, and $T$ has ...
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80 views

Mistake in integrating the inverse of a function?

The problem I have is: $$\int_0^{\frac1{\sqrt{3}}}\sqrt{x+\sqrt{x^2+1}}dx$$ I'm not asking how to solve it, I'm asking if anyone can point out to me the error that I have made: By differentiating ...
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63 views

Fastest way to find modular multiplicative inverse

I am looking for a fast way to find the modular multiplicate inverse of an integer $a$ in mod $p$. I am mainly interested in ...
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1answer
48 views

inverse of a tridiagonal matrix

Let $${A_{n \times n}} = \left[ {\begin{array}{*{20}{c}} {-2}&{1}&{}&{}&{}\\ {1}&{-2}&{1}&{}&{}\\ {}&{1}&{\ddots}&{\ddots}&{}\\ ...
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2answers
62 views

Finding inverse of a composite function

Let $f (x) = x^{3}+x$ and $g (x) =x^{3} -x$ for all x. I have to find derivative of $g\circ f^{-1}$ at $x=2$. My textbook did this: $(g \circ f^{-1})' (2) = \lim \limits_{h \to 0} \dfrac{g \circ ...
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0answers
10 views

Efficient inverses of many related matrices [duplicate]

Say I have a $N$-by-$N$ positive definite real matrix $\Sigma$ and I wish to compute the inverses (or equivalently Cholesky decompositions) of $(\Sigma + a_k I)^{-1}$ for a set of $K$ positive $a_k$. ...
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1answer
39 views

Finishing a proof: $f$ is injective if and only if it has a left inverse

I've already done a lot of searching (in particular: https://www.proofwiki.org/wiki/Injection_iff_Left_Inverse) to try to prove this statement: $f: A \to B$ is injective if and only if it has a ...