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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Finding inverse of a $3\times 4$ or $4\times 3$ matrix

Now I have no problem getting an inverse of a square matrix where you just calculate the matrix of minors, then apply matrix of co-factors and then transpose that and what you get you multiply by the ...
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1k views

An ill-conditioned matrix

If C is an ill-conditioned matrix and I want to get the inverse, one way is to take a pseudo-inverse of some sort. Instead, is the following, which uses the (normal) inverse, also a way to deal with ...
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55 views

If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$

Consider the statement: If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$. My book tells me this is suppose to be false, but I don't understand why. We know: If $f:X\to Y$ has ...
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151 views

Determine the greatest interval where the function is invertible

The assingment is to determine the greatest interval around $x=0$ where the function: $$f(x)=x^5-5x+3$$ is invertible. After that, determine $(f^{-1})'(3)$ I have totally forgotten all about ...
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1answer
45 views

The rank of general inverse of $A$ times $A$?

Supposing $X$ is the general inverse of $A$, that $AXA = A$. Then $XA$ is idempotent, that is $(XA)(XA) = XA$. Why is the rank of $XA$ equal to the rank of $A$ ? Thanks.
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Given the product of a unitary matrix and an orthogonal matrix, can it be easily inverted _without_ knowing these factors?

Given the product $M$ of a unitary matrix $U$ (i.e. $U^\dagger U=1$) and an orthogonal matrix $O$ (i.e. $O^TO=1$), can it be easily inverted without knowing $U$ and $O$? Sure enough, if $M=UO$, then ...
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137 views

How to show $AB^{-1}A=A$

Let $$A^{n \times n}=\begin{pmatrix} a & b &b & \dots & b \\ b & a &b & \dots & b \\ b & b & a & \dots & b \\ \vdots & \vdots & \vdots & ...
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1answer
92 views

Find whether or not an inverse exists algebraically

Is there an algebraic(without graphs) way to determine the existence of a function's inverse without using calculus? I'm an undergrad engineer and can obviously solve this using basic calculus, but ...
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89 views

Proof of Matrix Norm (Inverse Matrix)

Show for any induced matrix norm and nonsingular matrix A that $$ \left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1} $$ where $$ \left\|A^{-1}\right\| = ...
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126 views

Finding inverse of a matrix

This question is in my assignment. We are not allowed to use any symbol to represent any elementary row and column operations used in the solution. We must solve it step-by-step. Please help me to ...
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92 views

A bijective mapping from $\mathbb N^k$ to $\mathbb N$?

Having $k$ numbers $N_i\in\mathbb{N}$, I'm looking for a bijective mapping $f:\mathbb{N}\times\ldots\times\mathbb{N}\rightarrow\mathbb{N}$ So that ...
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154 views

Proof with functions and inverse - Spivak

How does he know that $f^{-1}$ is one-one? Doesn't he have to prove that? Or is he applying his first theorem in the chapter to $f$? That is $f$ is a function if and only if $f^{-1}$ is ...
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1answer
51 views

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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176 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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113 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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54 views

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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99 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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85 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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505 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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188 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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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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307 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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770 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
255 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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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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175 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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1answer
39 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
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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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47 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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163 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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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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114 views

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
75 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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146 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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1answer
76 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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257 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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1answer
113 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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Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$ [closed]

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots ...
3
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1answer
67 views

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?
3
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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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162 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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956 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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67 views

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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1answer
175 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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1answer
247 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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44 views

Closed form of the inverse of a function

Does anyone know what the analytic form of the inverse of $f(x)=e^x+x$? Thanks in advance
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61 views

Inverse of $x^2+\log^2\cos x$

I'm looking for the inverse of $$f(x)=x^2+(\log\cos x)^2$$ Where $f$ is defined from $[0,\pi/2)$ It dosen't have to be closed form, a sum, an integral or some special functions would be of interest ...
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1answer
70 views

Is there a closed form for the inverse of $y=x^{x^x}$?

It's pretty well known, and easy to derive, that $y=x^x$ has the inverse $y=\frac{\ln x}{W(\ln x)}$. I've had no luck trying to work out the inverse of any larger power towers, though. Is there any ...