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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If $A$ is an $n \times n$ matrix such that $A^3 = O_{3}$, show that $I - A$ is invertible with inverse $I + A + A^2$

So this question is basically a proof. If $A$ is an $n \times n$ matrix (so square) which satisfies the condition $A^3 = O_{3}$ ($A^{3}$ gives the $3 \times 3$ zero matrix), then show that $(I - A)$ ...
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To invert a Matrix, Condition number should be less than what?

I see that there is a matlab tag in this site, so I ask my question here and not in stackoverflow although it is also related to programming in matlab. I am going to invert a positive definite matrix ...
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164 views

Invertible Derivative

I'm trying to brush up on some differential geometry, but there's a subtle point I don't understand. Suppose $h$ is a diffeomorphism. Then the lecture notes here suggest that it's derivative $df_x$ is ...
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107 views

Formula for Nth Derivative of Matrix Inverse

I was looking for an equation for the nth derivative of a matrix inverse, ie $\frac{d^n \bf{A}^{-1}}{dx^n}$ I know that the first derivative $\frac{\text{d} \bf{A}^{-1}}{\text{d}x} = -\bf{A}^{-1} ...
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Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?

Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how? This seems like an easy question, but ...
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95 views

Inverse of a polynomial function

I want to find the inverse of $f(x)=\frac{3}{4}x^2-\frac{1}{4}x^3 $ when $0<x<2$. According to wolfram the answer is inverse I would like to know how can I find wolfram's inverse.
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53 views

Linear Algebra determinant and rank relation

True or False? If the determinant of a $4 \times 4$ matrix $A$ is $4$ then its rank must be $4$. Is it false or true? My guess is true, because the matrix $A$ is invertible. But there is ...
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82 views

does invertibility of product imply invertibility of each term of product?

Suppose $\mathcal{H}$ is a Hilbert space and the product $T_1T_2 \in B(\mathcal{H})$ is invertible. Does this imply that both $T_1, T_2$ are invertible ? I am unable to prove this since, unlike the ...
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469 views

Matrix Pseudo-Inverse using LU Decomposition?

What is the step by step numerical approach to calculate the pseudo-inverse of a matrix with M rows and N columns, using LU decomposition? So far, I have found this, but it uses singular value ...
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181 views

Dense pre-images implies continuous right inverse?

Suppose $f : \mathbb R \to \mathbb R$ is such that pre-image of every point under $f$ is dense in $\mathbb R$. This, of course, implies that $f$ is surjective, and hence has a right inverse ...
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34 views

PID question in Ireland and Rosen

Context: In Ireland and Rosen's 'A classic introduction to number theory' on page 11, the proof that in a PID$=R$, there is an integer $n$ such that, for a prime $p$ and any $b\in R$, $p^n \mid b , ...
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219 views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
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587 views

Calculating Moore-Penrose pseudo inverse

I have a problem with a project requiring me to calculate the Moore-Penrose pseudo inverse. I've also posted about this on StackOverflow, where you can see my progress. From what I understand from ...
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1answer
216 views

Inverse of $(A + B)$ and $(A + BCD)$?

Consider $A$ as an arbitrary matrix and $B$ as a symmetric matrix. Since $B$ is symmetric, therefore, it can be written as a $\Gamma \Delta \Gamma'$, where $\Delta$ is a diagonal matrix with ...
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1answer
71 views

Why $ g(p) = 0.5 p^{-0.2} + 0.5 p^{-0.5} $ has a well-defined inverse that is continuous and strictly decreasing.

A book that I am reading claims the following about the function $ g(p) = 0.5 p^{-0.2} + 0.5 p^{-0.5} $ (which is a demand function): Formal arguments based on the Intermediate Value Theorem and ...
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173 views

Need help finding inverse under $a\circ b = a^b b^a$

I'm going through some problems in Theorems, Corollaries, Lemmas, and Methods of Proof, and I'm stuck at a certain problem that seemed very interesting until I couldn't solve it for the life of me. ...
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37 views

Calculate the product ST, and infer from it the inverse of T.

S=\begin{pmatrix} 1/2 & 1/2 & 0\\ 1 & 0 & 0\\ -3/2 & 0 & 1/2 \end{pmatrix} T= \begin{pmatrix} 0 & 1 & 0\\ 2 & -1 & 4\\ 0 & 3 & 2 \end{pmatrix} I ...
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33 views

Inverse of Cartan matrix

The Cartan matrix of the root system $A_n$ looks like, denote it by $A'_n$ $$A'_n= \begin{bmatrix} 2 & -1 & 0 & 0&\ldots & 0 \\[0.3em] -1 & 2 & -1 ...
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131 views

Conditions for a matrix to be invertible

Let $n \geq m$ and let $C$ be a $n \times m$ full rank matrix, that is $rank(C) =m$. Considering that $D$ is a diagonal positive semidefinite matrix, under which conditions is the $ m \times m$ matrix ...
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48 views

Yet another inverse function to calculate

Is it possible to evaluate the inverse of this function, in order to obtain for each $y\in\mathbb R^+$ an explicit value of $f^{-1}(y)$? Thanks in advance! ...
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74 views

Is there a name for an algebraic structure like this?

I'm self studying abstract algebra. I see that in rings there's no requirement for a multiplicative inverse. Is there something similar except with no requirement for an additive inverse. For ...
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136 views

Definite integral - Please point to me my mistake

This emerged while I was investigating this question, i.e. the solution to the definite integral $$I_x = \int_0^\infty\left(5x^5+x\right)\operatorname{erfc}\left(x^5+x\right)\,dx$$ In a comment, its ...
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106 views

A formula for n-derivative of the inverse of a function?

Let $y=f^{-1}(x)$. As we know: \begin{align} \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{{f}'(y)} \end{align} Thereof we have: \begin{align} \frac{\mathrm{d^2} y}{\mathrm{d} ...
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Right Inverse for Surjective Function

Prove that if $f:X\to Y$ is a surjective function between sets, then there must exist a function $g:Y\rightarrow X$ such that $f\circ g=1_Y$. I know that the identity function is onto, and if $f$ ...
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74 views

How to compute $\text{trace}((A+D)^{-1}A)$

Give a diagonal perturbation matrix $D$ (which is not an identity matrix), is there a simple way to compute $$\text{trace}((A+D)^{-1}A)$$ Or is there a good approximation?
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243 views

Finding the inverse function, is there a technique?

I came across a way to find whether some number is inside a sequence of numbers. For example the sequence (simple function for positive odd numbers): $$a(n) = 2n + 1.$$ So the numbers inside it go: ...
3
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110 views

For $T$ compact, $I-T$ left or right invertible implies $I-T$ invertible

Let $S\in B(X)$ be a bounded linear operator from $X$ onto $X$ and let $T\in K(X)$ be a compact linear operator from $X$ onto $X$. Then $$ S(I-T)=I \iff (I-T)S=I. $$ I don't know if we need the fact ...
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55 views

Inverse Function Differential Equation [duplicate]

For the differential equation $$\frac{d}{dx}[y(x)]=y^{(-1)}(x)$$ where $y^{(-1)}(x)$ is the inverse of $y(x)$, find y(x). I gave up on finding the solution analytically pretty quickly and decided ...
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Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as with SVD?

Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as using SVD?
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What is the proper way to find the inverse of a function?

I am a little confused on the subject of inverse functions and the methods used to do the transformation from function to inverse. How do you make an inverse? Just so i can avoid any ambiguity in my ...
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1answer
137 views

Inverse of Ulam's spiral

I have a program and I need a function that takes a coordinate as input and returns an integer corresponding to the position in Ulam's spiral. The simple (but slow) way to do this would be to ...
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671 views

A square matrix A is invertible if and only if det A ≠ 0. Use the theorem above to find all values of k for which A is invertible

$$\begin{pmatrix} k & k & 0 \\ k^2 & 25 & k^2 \\ 0 & k & k \end{pmatrix}?$$ I did a sample question before this one: $$\begin{pmatrix} k & k & 0 \\ k^2 & 16 & ...
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How to take one's complement of a positive integer?

Of course we can do that by converting the number to binary and then converting it back to decimal, but to do that directly in decimal?
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Evaluate the derivative of an inverse function by using a table of values?

Given the function and derivative values in the table below, evaluate $\frac{d}{dx}f^{-1}(3)$ ...
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160 views

Pseudo inverse not equal inverse — conditions?

What are the conditions under which a the pseudo-inverse of a matrix is not equal to its inverse? I have a matrix equation: $$ AXB = C $$ which according to Laub (13.14, 13.15) has a solution if ...
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235 views

Non-negative matrix and inverse

Lately, I´ve been struggling with math homework and came across a question I´m not sure how to answer. I will be glad for any help... Suppose we have matrix $A$ (size $n\times n$) and its inverse ...
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Moore-Penrose Pseudo-inverse of a matrix on adding 1 new row/column

Given that I know the pseudo-inverse of a matrix(not necessarily a square matrix), how to calculate the pseudo-inverse of the matrix I get by adding a single row/column to the original matrix? i.e, ...
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38 views

The inverse of a transpose matrix to “cancel” the transpose?

When it comes to solving and equation containing matrices I don't always understand some of the rules involved. In particular, I am trying to figure out the derivation of the Gauss-Newton algorithm. ...
3
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1answer
25 views

Linear algebra proof that AB = On with A invertible only if B = On

$A,B \in Mn(R)$ so that $AB=0n$ and $A$ is an invertible matrix. Proof that $B=0n$ by definition $A$ is invertible so: $\exists C \in Mn : AC=CA=In$ so $A \ne 0n$ Then $AB=0n$ if $B=0n$ Here I can ...
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Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
3
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109 views

How find this matrix the inverse $A^{-1}$

Let $a,b>0$,and the matrix $A_{n\times n}$ and such $$A=\begin{bmatrix} a&b&0&\cdots&0&0\\ b&a&b&\cdots&0&0\\ 0&b&a&\cdots&0&0\\ ...
3
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2answers
155 views

convexity of inverse function

I have a question on the reverse of a convex function. Let $f(x)$ be a convex function. Is the reverse function, say $g(x)=f(x)^{-1}$, is necessarily a concave function ? Considering that such ...
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Inverse of identity plus scalar multiple of matrix

Given the matrix $M = ( I + \alpha D P )$, where $I$ is the nxn identity, $D$ is nxn symmetric and invertible, $P$ is nxn symmetric but not always invertible, and $\alpha$ is a scalar, is there a ...
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80 views

An invertible matrix

Given Matrix $A$, checking that its diagonal elements are nonzero or whether its determinant is nonzero, can we say the matrix is invertible for sure? Are there other properties that by looking at the ...
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380 views

Left inverse iff injective; right inverse iff surjective

For a function $f:A\to B$, the function $g:B\to A$ is called: a left inverse for $f$ if $g\circ f$ is the identity on $A$ (i.e., $g\circ f = {\rm id}_A$); and a right inverse for $f$ if ...
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Finding the number of the real roots of $a^x=g(x)$ where $g(x)$ is the inverse function of $f(x)=a^x$

Question : Let $a$ be a constant which satisfies $0\lt a\lt 1$. Letting $g(x)$ be the inverse function of $f(x)=a^x$, then find the number $N$ of the real roots of $f(x)=g(x)$. Motivation : This is ...
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84 views

Inverse of a sub-matrix

I have a multivariate Gaussian distribution with known $\mu$ and $\Sigma$. I want to evaluate it given a vector $x$. However, some of the attributes of this vector may be unknown, in which case I want ...
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54 views

Equation between the two branches of the lambert w function

Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$? For example the two square roots $r_1(y)$ and $r_2(y)$ of the equation ...
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125 views

Sparse matrix inverse multiplied by sparse matrices

I have the equation $\bf E = Y D^{-1} Y^\top$. $\bf D$ is a potentially large sparse $m \times m$ matrix, and $\bf Y$ is a sparse $n \times m$ matrix, where $n \ll m$. Is there a particularly ...
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588 views

How to find the inverse of the function?

$$f(x)=\frac{x+2}{5x-1}$$ Answer: $$f^{-1}(x)=\frac{x+2}{5x-1}$$ Can one of you explain how the inverse is the same exact thing as the original equation?