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Questions tagged [inverse]

Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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3answers
33 views

Calculating inverse trig expressions like cos(arctan -2)

I have some problems "connecting dots". All feedback is welcomed and really, really helpful! :) Task 1: calculate $\quad \tan{(\arcsin{(-\frac{3}{4}}))}$ Solution: $\tan{(\arcsin{-\frac{3}{4}})} = ...
0
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0answers
7 views

Inverting a function from asymptotic expansion

Can I invert the following functions to obtain $r(\rho)$? $\rho=r+a+b r^{q}$, where $q<0$ $\rho=cr+dr^{q}$, where $q>0$ $\rho=r+\frac{b_{0}}{2}\left(-1+\ln[2]-\ln[b_{0}/2]-\ln[1/r]\right)$ ...
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1answer
62 views

Given a matrix A such that $a_{ij} = \frac{1}{i + j - 1}$, prove that A is invertible and that $A^{-1}$ has all integer entries. [duplicate]

Essentially just the title. This is supposed to be the introductory Upper Division Linear Algebra class so we haven't covered the determinant yet. I'm mentioning this because I am aware of the theorem ...
0
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1answer
19 views

Find all units in the ring Z[i] = { a+bi : a,b ϵ Z } [duplicate]

Find all units in the ring $ Z[i] $= { $a+bi$ : $a,b$ ϵ $Z$ }. I faced a similar problem to find all the invertible matrices in $Z$. I concluded the solution must be all matrices of det ($\pm1$). I ...
4
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1answer
44 views

Compact embedding of the domain and compact inverse

I have several problems in showing this point of a problem: we consider $X$ Banach space and $T: D(T) \to X$ a closed operator with domain $D(T) \subseteq X$. Let be $T$ bounded, invertible and ...
2
votes
1answer
28 views

Which one is most cost expensive to solve a linear equation? LU or inverse?

Which one is the most expensive way to solve for linear equation? LU-decomposition $$A = LU$$ Or finding the inverse $$A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A)$$ If I have to choose, I ...
2
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4answers
49 views

What is the intuition behind why a rank deficient matrix does not have an inverse?

Suppose that we have a $p$ dimensional square matrix $A$ whose rank is less than $p$. We know that such a matrix cannot have an inverse and there are several different ways to prove that the $A$ does ...
2
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0answers
64 views

How do I invert this matrix?

Given two vectors $$\vec{v} = \begin{pmatrix} v_1\\ \vdots\\ v_n \end{pmatrix} , \vec{w} = \begin{pmatrix} w_1\\ \vdots\\ w_n \end{pmatrix} \in \mathbb{R}^n$$ such that ...
3
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1answer
43 views

If $g$ is the inverse of function $f$ and $f'(x)= \frac{1}{1+x^n}$, Find $g'(x)$

I tried the question and got an answer by the following steps: $f(g(x))=x$ Differentiating both sides w.r.t to $x$, we get $f'(g(x)).g'(x)=1$ And therefore, $g'(x)=1+\left[{g(x)} \right]^n$ Now ...
6
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1answer
71 views

Fast way to Invert ADA' when D is a diagonal matrix that changes each iteration?

So I have a statistical learning algorithm in which D is a diagonal matrix that changes each iteration while A stays the same. I'm looking for a fast way to invert ADA' each iteration which ends up ...
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2answers
47 views

Matrix equations, simplify them.

I have some equations and I don't know am I doing the simplification right. Can someone check it? For example, we have an equation $AX = 4X + B$, where $A$ and $B$ are matrices. So, what I have done ...
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0answers
14 views

Inverse of matrix expansion with negative exponents

The answer to this question shows that if I have a real nonsingular matrix $M$, such that its Taylor expansion in $\epsilon$ is $$M(x+\epsilon)= \sum_{n=0}^\infty M_n(x) \epsilon^n $$ its inverse ...
0
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1answer
32 views

Reverse probability problem.

Suppose there are two cards A and B. Both of the cards have a yellow side and a green side. When tossed in the air the probability of the yellow side facing up is %31 for Card A and 35% for Card B. A ...
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2answers
37 views

Inverse of identity minus matrix exponential

I am trying to analytically find the inverse of a matrix given by: \begin{align} W = \left( I - \alpha e^A \right)^{-1}, \end{align} where $I$ is the identity matrix of appropriate size, $e^A$ ...
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2answers
35 views

how to find the inverse

gamma using this equation $1-\sqrt{1-x^2/c^2}$ where c = 1 and x= 0.0 - 1.0 the speed of c for example $1-\sqrt{1-.886^2/1^2}$ = y = 0.5363147619 gives me the y values on the graph. How do I ...
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0answers
28 views

Help with this matrix problem about the inverse of a particular matrix

Is the following matrix invertible? $ \begin{bmatrix} x & a & a & \dots & a \\ a & x & a & \dots & a \\ a & a & x & \dots & a \\ \vdots & \vdots &...
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1answer
46 views

What is the best algorithm to find the inverse of matrix $A$

I'm going to build a C-library so I can do linear algebra at embedded systems. It's most for machine learning. https://github.com/DanielMartensson/EmbeddedAlgebra/ Anyway, I need to compute inverse ...
1
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1answer
43 views

Inverting a power series matrix

Let's assume I want to invert a matrix function $M=M(x)$, which is expressed as a power series of the small parameter $\epsilon$ $$M = M_0(x) + M_1(x) \epsilon + M_2(x) \epsilon^2 + \mathcal{O}(\...
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0answers
15 views

Perturbing a matrix with asymptotic constraints

I want to expand the following function in series of $\epsilon = a/l$: $$f=\frac{h \cos\alpha + l\sin\alpha}{l \cos \alpha + c \sin \alpha}$$ I know from the physics of the problem that also $$\frac{...
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1answer
44 views

Given a Fibonacci number, find what number in the sequence it is [duplicate]

I came across the formula for the $n$th Fibonacci number: $$\frac{\Phi^n-(-\Phi)^n}{\sqrt5} = x,$$ where $x$ is the $n$th Fibonacci number. This formula works one way around, but I cannot seem to ...
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1answer
23 views

Characterization of invertible functions

Lemma I have managed to prove first part that f is bijection. Attempt I'm stuck on proving that $g=h=f^{-1}$ I tried the set theoretic approach, trying to show that the $2$ functions $g$ and $f^{-...
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0answers
22 views

How to reverse values in a reciprocal

How do i reverse the values of a reciprocal so that at 0 the speed of light the reciprocal is 0.0 and near at the speed of light the reciprocal is 1. Basically I want to reverse the values in the ...
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2answers
34 views

the inverse of a sum of two symmetric for schur completion?

I have a up-triangulate Jacobi matrix J which can be blocked like : $J = \begin{bmatrix}A & B\\ 0 & C\end{bmatrix} $ both A and C are up-triangulate, we can get Hessian matrix H by: $H = J'...
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1answer
26 views

Trick for Inverse Hollow Matrix Calculation (Self-Answered)

Let $A$ be the hollow matrix : $$ A=\begin{pmatrix} 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0 \end{pmatrix} $$ Find the inverse matrix $A^{-1}$ without ...
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2answers
38 views

Inverse of matrix product

Let A be an $n\times n$ matrix and B be an $n\times m$ matrix with $m<n$ can I use this identity? $(AB)^{+}=B^+A^{-1}$ I am not sure that this is the right inverse of the product $AB$ if this is ...
3
votes
2answers
91 views

Invert a $4 \times 4$ matrix with a given structure?

When tryig to fit $f(x,y) = a+bx+cy+dxy$ to the values of four points, we will have to invert following matrix. (Let us assume that $x_i,y_i$ are chosen suitably such that it is regular.) We could ...
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votes
1answer
61 views

Is there a generalized way to find an inverse of a function defined as a sum (if the inverse exists)?

Suppose I had a summation for a monotonic increasing function, like $f(n)=\sum_{k=1}^{n}k$. We already know this has a closed form solution in the form of $\frac{n(n+1)}{2}$. If you want to find $n$, ...
2
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2answers
47 views

Proof that $\exists x \in C$ that $A - x\cdot I_n$ is invertible

We know that $A \in \mathbb C^{n,n}$ Proof that $\exists x \in C$ that $A - x\cdot I_n$ is invertible. How many such x exists? I am thinking about this problem and generally I have only few little ...
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1answer
21 views

Left and right inverses of functions.

Let $X$ be the set of real transcendental numbers. Define the realtion $\sim$ on $X$ by $x\sim y$ iff $x-y \in \mathbb{Q}$ is an equivalence relation. Let $Y$ denote the set of equivalence classes ...
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1answer
33 views

inverse of polynomial as sum of two power series

given polynomial $$g(x) =ax^2+bx+c$$ I try to find $g(x)^{-1}$ as a sum of two power series. I wrote $g(x) = \alpha\beta(1-\frac{x}{\alpha})(1-\frac{x}{\alpha})$ when $ \alpha,\beta = \frac{ - b\pm\...
3
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0answers
222 views

Woodbury Matrix Inversion

I am trying to invert a matrix using Woodbury identity. The inversion using Cholesky decomposition has the following pseudo-code: For $t=1,2,...$ $(1)\;\; \text{Read}\;x_t\in\mathbb{R}^n$ ...
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1answer
32 views

Setting up the standard matrix for orthogonal projection

For a subspace $V$ of $\mathbb{R}^4$, you are given these three ordered bases: $A= (\mathbf{a}=(1,-2,-1,3), \mathbf{b}=(1,3,-2,-4))$ $B= (\mathbf{a}=(1,-2,-1,3), \mathbf{c}=(2,1,-3,-1))$ ...
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1answer
27 views

Blockwise Matrix Inversion Stability

I have implemented a blockwise matrix inversion. When comparing the blockwise inversion to an inverse of the entire matrix I am seeing deviations from the correct inverse. What should the expectation ...
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1answer
21 views

How to calculate the inverse of $\mathbb{R}^2$-version of the inverse Cayley map

As we know the inverse Cayley map can be expressed as $f(z)=i\frac{z+1}{1-z}$, i.e. a biholomorphism from the complex unit disk to the upper half complex plane. I have algebraically rewritten this ...
2
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0answers
82 views

Inverse of identity minus exponential matrix

I am trying to analytically find the inverse of a matrix given by: \begin{align} W = \left( I - \alpha e^A \right)^{-1}, \end{align} where $I$ is the identity matrix of appropriate size, $e^A$ ...
3
votes
0answers
57 views

Calculating the diagonal of $(I-Q)^{-1}$ efficiently

Motivation: I'm trying to write code to solve an equation efficiently. Directly calculating the result is easy, but involves matrix inversions that consume an impractical amount of memory at the scale ...
2
votes
2answers
42 views

True or false statements about a square matrix

Consider the following four statements about an $n \times n$ matrix $A$. $(i)$ If $det(A) \neq 0$, then $A$ is a product of elementary matrices. $(ii)$ The equation $Ax=b$ can be solved ...
2
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3answers
96 views

Why don't similar matrices have same eigenvectors and eigenvalues?

What is wrong with this proof: Suppose R and T are similar operators and R has eigenvalue 2 for some eigenvector $v$. By property of similar matrices: $R=STS^{-1}$ Therefore: $Rv = STS^{-1}v$ $2v ...
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votes
4answers
61 views

Is the action of taking the inverse distributive in any way? For example, is $(A+B)^{-1}= A^{-1} + B^{-1}?$

Sorry for not formatting properly, I can't seem to get the exponents to show up properly (ex. $A^{-1}$). Can you distribute the act of taking the inverse over a pair of brackets? For example, is $...
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0answers
8 views

Inversion of Gradient Noise Function

Brief As Possible Given any equation $f(x)$ which adheres to a predefined pattern, there is always a way (sometimes not directly calculable) to get $f^{-1}(x)$ (which may be a set) such that $f^{-1}(...
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0answers
22 views

Gauss Jordan complex matrix inversion - Choice of pivot element

When inverting a matrix of consisting of complex numbers, how is the pivot element chosen? In a real matrix the smallest number is chosen, how can the choice be made when the elements are complex? ...
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1answer
38 views

Inverse of function and derivative

The problem says: If $f(x)=\frac{4x^3}{(x^2+1)}$ find $(f^{-1})'(2)$. I can show that the function is one-to-one and maybe I should use $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ but I dont know how. ...
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1answer
23 views

Invertible matrix properties of a matrix

I have here the following question: Let $X$ be the $5 \times 5$ matrix "full of ones": $X = \begin{pmatrix}1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 &...
2
votes
2answers
76 views

Show that if $(A+2I)^2=0$, then $A+\lambda I$ is invertible for $\lambda \ne 2$.

Show that if $(A+2I)^2=0$, then $A+\lambda I$ is invertible for $\lambda \ne 2$. I tried to solve this by treating $(A+\lambda I)v=0$ as linear equation system, and proving that $v$ must be $0$ (...
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0answers
15 views

matlab - Optimization with inverse and pseudoinverse

Let us assume I have to optimize this system: $$\min_{x\in S} \left|\left|\left(E\begin{bmatrix} I_n\\ A'(x)^{-1}C'(x)\\ \end{bmatrix}\right)^+ a -b\right|\right|^{2}$$ Where x is the vector ...
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0answers
23 views

What is the largest condition number a 64-bit computer can take to do matrix inversion to give good result?

In my problem, it looks like $10^{13}$ is a red-line, once it crosses, the performance of matrix inversion goes down. But why? Or do you have better idea for that?
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3answers
44 views

Let $A,B,C,D \in \mathbb{R}^{n×n}$. Show that if $A, C, B−AC^{−1}D,$ and $D−CA^{−1}B$ are nonsingular then that the following matrix has the matrix:

Let $A,B,C,D \in \mathbb{R}^{n×n}$. Show that if $A, C, B−AC^{−1}D,$ and $D−CA^{−1}B$ are nonsingular then $\left[ \begin{smallmatrix} A&B\\ C&D \end{smallmatrix} \right]^{-1} = \left[ \begin{...
2
votes
2answers
63 views

Find the additive inverse of binary number

My online assembly class doesn't really show us how to find the additive inverse of finding the additive inverse of binary, and I can't find much online. The question is: find the additive inverse of ...
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votes
1answer
33 views

Finding inverse to elements

I am trying to find the inverse of the following elements $2+\sqrt7$, $\sqrt3-\sqrt2$ and $1+i\sqrt5$. I would be very much thankful if someone could help me with this one.
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votes
1answer
37 views

Find all the bijective functions $f:[0,1]\to[0,1]$ such that $x=\frac{1}{2}\big(f(x)+f^{-1}(x)\big)$ for all $x\in[0,1]$.

Find all bijective functions $ f : [0,1] \to [0,1]$ that satisfy the equation $$x=\frac{1}{2} \big(f(x) +f^{-1} (x)\big)\,\forall x \in[0,1]\,.$$ I honestly don't know how to approach this. ...