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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2
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1answer
31 views

Row sum of inverse of a matrix

Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix} $$ All the elements of A are ...
1
vote
1answer
24 views

Inverse modulo certain numbers.

Let $b,n$ be given positive integers. I would like to give some formula for the multiplicative inverse $a^*$ of a positive integer $a$ to a modulo of the form $4ab-n$, where $(a,n)=1$, i.e. $$ aa^*\...
3
votes
1answer
129 views

Inverse of a matrix and its transpose

I'm trying to figure out why the calculation below works. I do know that $(A^T)^{-1} = (A^{-1})^T$. The matrix A = $\begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1 ...
1
vote
1answer
31 views

Right inverse matrix

I know that if $A, B$ and $C$ are square matrices such that $$ AC=I \quad \mbox{and} \quad BA=I, $$ then \begin{eqnarray*} AC=I & \Rightarrow & BAC=B\\ & \Rightarrow &IC=B\\ & \...
2
votes
1answer
58 views

How can I divide a vector by a matrix?

I am trying to go backwards through a neural network. I have an output and I want to see what input would lead to that output. To go forwards I start with a vector and multiply by a matrix and then ...
0
votes
1answer
37 views

How can I solve an equation based off of a quadrant and equation form given an angle?

Given that $\dfrac{3\pi}{2} < z < 2\pi$ $x = arccos(sin(z))$ Given different values for z (which are angles on the unit circle) how would I write the results in these two forms, where C is a ...
5
votes
1answer
789 views

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 ...
0
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5answers
48 views

Inverse Equation of the Given Equation

Having a bit of a problem getting the inverse of the following equation: $$f(x) = \sqrt{9-x^2}$$ I had an answer which was equal to $3-x$ but when I used sites like Mathway and Wolfram to check my ...
0
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1answer
19 views

Inverse Parameters of a Pan-Tilt Rotation Possible?

I have a 2-parameter (tilt,pan) rotation computed as tilt followed by pan, i.e. two rotation matrices multiplied together: $$R(t,p)=\begin{pmatrix} c_p & s_p s_t & s_p c_t \\ 0 & c_t &...
4
votes
1answer
433 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 $T$ ...
2
votes
1answer
184 views

is this matrix invertible

Is the following matrix invertible? $\left[ \begin{matrix} \sum\limits_{x=1}^{n}{1} & \sum\limits_{x=1}^{n}{x} & \sum\limits_{x=1}^{n}{{{x}^{2}}} & \cdots & \sum\limits_{x=1}^{n}{{...
1
vote
2answers
406 views

Inverse of the Modified Bessel function

Is there any chance of having a formula or approximation to inverse the Modified Bessel function of the first kind? I mean to solve $I_M(x)$ with respect to x: $I^{-1}_M(x)$? Thanks in advance
0
votes
0answers
24 views

Linear regression of matrix elements to get the minimal polynomial to perform a matrix inversion?

So each matrix $\bf A$ fulfils an equation for it's minimal polynomial $P_m({\bf A})$: $$P_m({\bf A}) = 0 \Leftrightarrow \sum_{k=0}^{k_n}c_k{\bf A}^k = 0$$ We can by multiplying with $A^{-1}$ and ...
1
vote
3answers
42 views

Calculate multiplicative inverse of $95$ in group of order $n=101$ which is subgroup of $(\mathbb{F}_{607}^*,\cdot)$

In the notes where I'm studying from there is written: "Let $G=\langle g\rangle$ be a subgroup of $(\mathbb{F}_{607}^*,\cdot)$ with $g=64$ and order $n=101$" but that felt strange to me; since I know ...
0
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0answers
24 views

Solving for $f_2(x, y)$ in $\int_{\mathbb{R}}f_1(x, y) f_2(x, y) dx = g(y)$

Let me first present the problem in its most general form, and then work down to a specific case of interest and my motivation. Suppose $f_1: \mathbb{R}^2 \rightarrow \mathbb{R}_{\ge 0}$ and $g: \...
-1
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1answer
2k views

Inverse of binary matrix

I have tried creating an inverse of a binary matrix using the identity matrix method. Have an identity matrix alongside the square matrix and perform all the operations to convert the square matrix to ...
0
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1answer
13 views

Square Root of a matrix: transpose or inverse of eigen vectors?

Here is described that the square root of a matrix is defined as K^1/2 = V*D^1/2*V^-1 At the end of scetion 4 of this paper we can see W = K^-1/2es In the ...
1
vote
1answer
51 views

How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix?

How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix? Assume that $M$ is an $n \times m$ matrix with $n \leq m$. Is it possible to obtain ...
0
votes
0answers
9 views

Is there a simple algorithm to compute polynomial inverses over cyclotomic polynomials?

I'm working with polynomial inversions in a ring built over the nth-cyclotomic polynomial, with $n = 2^i$. As usual, I'm applying Extended Euclidean algorithm on this, an approach that does not scales ...
0
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0answers
42 views

$y=x/(1+a(x))$, $\quad$ $x=y/(1+b(y))$. What is known about $a\mapsto b$?

\begin{align} y & = f(x) = \frac x {\displaystyle 1 + \sum_{n=1}^\infty a_n \frac{x^n}{n!}} \\[15pt] x & = f^{-1}(y) = \frac y {\displaystyle 1 + \sum_{n=1}^\infty b_n \frac{y^n}{n!}} \end{...
2
votes
1answer
40 views

Is this how eigenvalues of some matrix $A$ are related to the inverse of $A$?

Let $A$ be an invertible $n\times n$ matrix. If $$Av = \lambda v \qquad (1)$$ for some $v$ and $\lambda$ then $\lambda$ is an eigenvalue of $A$ and $v$ a corresponding eigenvector. Equation $(1)$ may ...
0
votes
1answer
50 views

Inverse operation of tetration and how it is computed?

If $c=a+b$, then $a=c-b$ and $b=c-a$. If $c=a\times b$, then $a=\frac{c}{b}$ and $b=\frac{c}{a}$. If $c=a^b$, then $a = \sqrt [b]{c} =c^{\frac{1}{b}}$ and $b=log_ac$. What are the analogous inverse ...
0
votes
2answers
41 views

Is $\text{arccosec}(x) = \arcsin\left(\frac{1}{x}\right)$ for all $x \in ℝ?$

Is $\text{arccosec}(x) = \arcsin\left(\frac{1}{x}\right)$ for all $x \in ℝ?$ I'm still really new to trigonometric inverses, so if the above was cleared up I'd be grateful. Thanks.
2
votes
3answers
32 views

Complex inverse function

I've got a problem when solving an inverse function. Usually when I have a basic function and trying to find its inverse is not a problem. I just solve for X and find it. But now I've got a more ...
13
votes
5answers
16k views

What's the difference between arccos(x) and sec(x)

My question might sound dumb, but I don't really see why the graphics of arccos(x) and sec(x) are different, because as far as I know arccos is the inverse cosine function (cos(x)^-1) and sec equals 1/...
4
votes
2answers
126 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 ...
5
votes
1answer
57 views

Prove a matrix expression leads to an invertible matrix?

I want to prove matrix $C$ is invertible: $$C=I-A^TB(B^TB)^{-1}B^TA(A^TA)^{-1},$$ where $I$ is an identity matrix of appropriate dimensions, and $(A^TA)^{-1}$ and $(B^TB)^{-1}$ imply both $A$ and $B$ ...
1
vote
1answer
56 views

How to show the matrix $\left( \binom{x-i}{j-1}\right)_{1\leq i,j\leq 2r+1}$ has determinant (-1)^r and it's inverse?

After playing around in mathematica, I found that the matrix $\left( \binom{x-i}{j-1}\right)_{1\leq i,j\leq 2r+1}$ has determinant $(-1)^r$ for the first few $r$'s. How can I prove this this, or at ...
0
votes
2answers
44 views

Uniqueness of Inverse

I am having trouble understanding the logic for a few steps in the following. I'll point the steps out at the end. If B and C are inverses of a square matrix A, then B = C. Proof: Since B is an ...
0
votes
2answers
37 views

Finding the expression of the inverse of $(AB)^T$

I know that $(AB)^T$ = $B^TA^T$ and that $(A^T)^{-1}= (A^{-1})^T$ but couldn't reach any convincing answer. Can someone demonstrate the expression.
1
vote
1answer
44 views

What are all the Banach algebras where $\|a\|\|a^{-1}\|=1$? [closed]

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?
1
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1answer
37 views

Kernel of inverse of the Möbius transformation

Given $f(z)=\frac{az+b}{cz+d}$ the Möbius transformation. Calculate $ker(f^{-1}(id))$. $$f(z)=\frac{az+b}{cz+d}=w\implies w(cz+d)=az+b\implies z(wc-a)+wd-b=0$$$$\implies z=\frac{-wd+b}{wc-a}=f^{-1}(z)...
1
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2answers
25 views

How to normalize and inverse a vector so it sums to 1 ?

I understand how normalization works. You sum up the individual values of the vector, you divide each value by the sum, and voila... they sum to 1. Why doesn't it work when you subtract them from ...
1
vote
1answer
67 views

Calculating inverse function with 2 variables

$f: R^{2}\mapsto R^{2}$ $(x,y)\mapsto (x^{2}-4y^{2}+x, -xy+3y)$ I should calculate inverse function of $f$ in point $(3,1)$. I tried to do $(x,y)\mapsto(u,v)$, but I just dont know how to get x ...
0
votes
2answers
44 views

Repeated use of Woodbury formula

I want to calculate the $x$ dependency of $\left(I + A \Lambda (x) A^{T}+B\Omega(x)B^{T}\right)^{-1}$ explicitly, where $I$ is a $n\times n$ matrix. Here $\Lambda (x) $ and $\Omega(x)$ are diagonal $...
0
votes
0answers
34 views

Inverse of the sum of identiy matrix and a symmetric matrix

Is there a simple way to solve $(I + A) X = B$, where $I$ is the identity matrix, and $A$ is a symmetric matrix?
3
votes
3answers
98 views

Inverse of a matrix with $a+1$ on the diagonal and $a$ in other places

Let $a>0$. Let $A$ be the $n\times n$ matrix with $a+1$ on the diagonal and $a$ in all other entries. How can one compute $A^{-1}$ as a function of $n$?
1
vote
1answer
38 views

Inverse of a matrix with main diagonal elements approaching infinity

Let $A$ be a invertible, symmetric, positive definite $p \times p$ covariance matrix with main diagonal elements $a_{ii},~i = 1,~\ldots,~p$. If all main diagonal elements would approach $\infty$, ...
0
votes
2answers
759 views

Inverse functions and tangent line

Let $f(x) = \frac14x^3 + 12x + 6$ and let $y = f^{-1}(x)$ be the inverse function of $f$. Determine the $x$-coordinates of the two points on the graph of the inverse function where the tangent line is ...
0
votes
0answers
16 views

How to solve a LMI with inverse matrix and quadratic form

I have to solve the following LMI, where $\Sigma$ is a symmetric positive definite matrix. K,D and $\Sigma$ are unknown: $$\left[\begin{array}{cc} K\Sigma^{-1}K^{T}+DVD^{T}+I & KA^{T}\\ AK^{T} &...
1
vote
1answer
108 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? ...
0
votes
3answers
63 views

Inverse of $f(x) = 2x^2+8x+13?$

How can you find the inverse of $f(x) = 2x^2+8x+13?$ This is what I've tried so far: $y = 2x^2+8x+13$ $x = 2y^2+8y+13$ $x-13 = 2y^2+8y$ $x-13=y(y+8)$ This is where I got stuck. To be clear, I want ...
8
votes
4answers
672 views

is it true every left inverse of a matrix is also right inverse of it?

I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ ...
3
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0answers
57 views

Time complexity of inverting an $n \times n$ matrix which is the sum of a rank-$m$ matrix and a full-rank diagonal matrix

I want to know the time complexity of inverting $K$, where $K$ is an positive-definite $n\times n$ matrix: $$K=\Lambda+Q$$, where $\Lambda$ and $Q$ are both $n\times n$ matrix, $\Lambda$ is a full-...
0
votes
1answer
56 views

What is the inverse Laplace transform of $\lfloor s \rfloor$?

How can we find the inverse Laplace transform of: $[x]$ (floor function) ? My question isn't LLaplace transform of floor function i asked the "inverse" laplace transform of floor function $$\mathcal{...
1
vote
1answer
35 views

Show there exists $C\in\Bbb{R}^n$ such that $|C-A_i|=|B-A_i|+u_i$, with $A_i,B\in \Bbb{R}^n$ and $u_i$ close enough to $0$

Let $A_1,...,A_n,B$ be vectors in the $n$-dimensional Euclidean Space, such that they are never on the same affine $(n-1)$-dimensional subspace. (What? Is that a way to say they span $\Bbb{R}^n$?). ...
0
votes
1answer
51 views

How could I search the inverse operator $(Af)^{-1}(x)$

I am try to search $A^{-1}$ when I define $A:L^2[0,2] \rightarrow L^2 [0,2] $ when $$(Af)(x)=x^{-1/4}f (\sqrt {2x}) $$ What I do: I consider that $ (Af)^{-1}((Af)(x))=Ix=x \Longleftrightarrow (Af)^{...
0
votes
0answers
16 views

Conditional Mean Given Precision Matrix While Avoiding Inversions

I'm working on a problem where I need to compute a conditional mean directly from a precision matrix (the inverse of covariance matrix). Let $\boldsymbol \mu$ be a mean vector partitioned into $$\...
0
votes
0answers
23 views

Inverse projection matrix 2D to 3D

I am writing a simple computer vision application in which reports the position of coloured dots on the floor. The floor is observed by a camera for which I have the correct projection matrix. I.E. If ...
8
votes
4answers
19k views

Inverse function of a polynomial

What is the inverse function of $f(x) = x^5 + 2x^3 + x - 1?$ I have no idea how to find the inverse of a polynomial, so I would greatly appreciate it if someone could show me the steps to solving this ...