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.

learn more… | top users | synonyms

3
votes
2answers
74 views

a practical question about matrix derivative with inverse and chain rule: dimension mismatch

Recently, I was trying to take the following derivative $$ \dfrac{\partial (X^TV^{-1}X)^{-1}}{\partial V} $$ I was referring to matrix cookbook to solve it, where I found several useful equations: ...
1
vote
3answers
352 views

Why does Arccos(Sin(x)) look like this??

I can kind of understand the main direction (slope) of $y$ over the different $x$ intervals, but I can't figure out why the values of $y$ take on the shape of straight lines and not curves looking ...
0
votes
3answers
40 views

Consider $f : \mathbb{N} → \mathbb{Z}$ defined as $f (n) = \frac{(−1)^n (2n−1)+1}{4}$. Find its inverse.

I cannot find an inverse of this function for $f(n) = x$, where $x$ is an integer, that gives out a natural number. Some guidance would be very helpful... I already know the function is bijective so ...
0
votes
0answers
28 views

Hessian for inverse probit link

I'm trying to calculate Hessian and Fisher Information for binomial model using inverse probit link, Suppose likelihood function is $L(\pi)=\prod\limits_{i=1}^n \pi_i^{y_i}(1-\pi_i)^{1-y_i}$ and ...
0
votes
4answers
45 views

Inverse of function - Difficulty solving [closed]

$$y=\sqrt{35\tan (\frac{\pi }{180}x)}$$ I have a really hard time finding the inverse of this particular function. Can anyone shine some light on why that may be, or alternatively solve it if I'm ...
2
votes
1answer
62 views

Proving a matrix is invertible

There's a linear algebra problem I'm having some trouble with: Let $A$ and $B$ be square matrices with the dimensions $n\times n$. Prove or disprove: If $A^2 + BA$ is invertible, then $A$ is also ...
1
vote
2answers
54 views

Are these equiv? $\cot^{-1}(-x) = -\cot^{-1}(x)$

$-\cot(y)=x$ Let's say you want to put this in terms of y... $\cot(y)=-x$ $y = \cot^{-1}(-x)$ =================== Is this also valid ? $-\cot(y)=x$ $y = -\cot^{-1}(x)$ If yes, how exactly does ...
-1
votes
3answers
68 views

If the product of two square matrices is invertible, then both matrices are invertible

If $A$ and $B$ are $n\times n$ matrices, and $AB$ is invertible then $A$ and $B$ are invertible. I started out by writing that since $AB$ is invertible, then for the equation $ABx=b$ has a unique ...
1
vote
2answers
54 views

If f(x+y)=f(x)*f(y) and f is a bijection, show that its inverse satisfies this function equation

I'm having trouble with this problem. I'm not even sure how to go about finding the inverse of an equation with both x and y. Here is the problem: If $f(x+y)=f(x)*f(y)$ and $f$ is a bijection, show ...
1
vote
6answers
55 views

The multiplicative conjugate of an invertible matrix is invertible

If $A,B,C$ are $n \times n$ (real) matrices and $A$ and $B$ are invertible, with $AB=BC$, prove that $C$ is also invertible. My attempted proof is $(B^{-1})(AB) = (B^{-1})(BC)$. Then $A(B^{-1})(B) = ...
2
votes
1answer
40 views

Properties of adjoint matrix in a finite dimensional inner product space

let $V$ be a finite dimensional inner product space. Let $T$ be a linear operator on $V$. Prove that there exists an invertible linear operator $U$ such that $U^{-1}TT^*U = T^*T $ where $T^*$ is ...
0
votes
1answer
39 views

why inverse trigonometric function DNE

Determine the exact value $\arccos\left[\sec\left(\dfrac{7\pi}{6}\right)\right]$ and $\text{arcsec}\left[\sin\left(\dfrac{13\pi}{6}\right)\right]$ Why does the exact value of these two questions not ...
3
votes
2answers
75 views

How to find the second derivative of the inverse function of $f(x)$ at $x=0$?

If $f\displaystyle(x)=\int _{ \sin x }^{ \cos x }{ \frac { dt }{ e^{ t }\sqrt { 1-t^2 } } } $ where $x\in[0,\pi/2]$ then how to find the second derivative of the inverse function of $f(x)$ at $x=0$ ?...
1
vote
1answer
18 views

Calculating differential of inverse function.

trying to find $(f'^{-1})(a)$ and am getting the wrong answer.
0
votes
1answer
53 views

Finding an inverse matrix

I should find the inverse matrix of the matrix: \begin{pmatrix} n & n & \cdots & n & n \\ n-1 & n-1 & \cdots & n-1 & 0 \\ \vdots & & & & \...
0
votes
1answer
22 views

fine the inverse of $[2]$ and $[23]$ in$ \mathbb{Z}_{41}$

I know the inverse of [23] is [23] * [25] = 575 575 congruent to 1 mod 41 [25] is the inverse I have started the other one but I am doing something wrong I got [2] * [41] = [82] = [0] 82 ...
0
votes
1answer
42 views

If $f:\{1,2,3\}\to\{1,2,3\}$ is bijective and $f(1)=2$, can we verify that $f(2)=1$?

I know that if $f$ is bijective, if $f(1)=2$ then $f^{-1}(2)=1$ but if $f:\{1,2,3\}\rightarrow\{1,2,3\}$ then does it mean necessarily that $f(2)=1$?
0
votes
1answer
44 views

proof: inverse of lower triangular identity matrix

As you know that is enough negating below of diagonal to inverse of lower triangular identity matrix. example $$A = \left(\begin{matrix} 1 & 0 & 0 & 0 \\ 3 & 1 & ...
1
vote
1answer
50 views

The column space of $A^2$ is all of $\mathbb R^n$ if and only if the column space of $A$ is all of $\mathbb R^n$

How would I go about proving the following statement? "Let $A$ be an $n \times n$ matrix. $\operatorname{Col}(A^2)=\mathbb{R}^n$ if and only if $\operatorname{Col}(A)=\mathbb{R}^n$" I started off by ...
0
votes
0answers
34 views

how do I find the inverse function of $x = ky - \cos(y)$ where $k$ is a constant

In other words, I'd like to express $y$ in terms of $x$ instead of having $x$ expressed in terms of $y$. I just don't know how to proceed!
0
votes
2answers
39 views

Square of Elementary Matrix Proof

I'm having trouble proving the following statement: "There exists an elementary matrix $E_1$ such that $E_1^2 = I$" I'm thinking about how the inverse of $E_1$ is equal to $E_1$ (so $E_1^{-1} = E_1$)...
0
votes
0answers
22 views

Existence of an inverse function in this functional equation

Let $V,W$ each be connected and separable sets. Suppose we have continuous functions $F:\mathbb{R}\times V\rightarrow \mathbb{R}$, $G:\mathbb{R}\times V\rightarrow \mathbb{R}$, $f:V\times W\rightarrow ...
2
votes
0answers
97 views

Compact convergence of inverse functions

Consider two metric spaces $X$ and $Y$ and a sequence of functions $f_n\colon X\to Y$ together with a function $f\colon X\to Y$. Assume, all $f_n$ and $f$ have inverse functions $g_n$ and $g$, say. It ...
2
votes
0answers
33 views

Continuity at an inner point of an interval implies the continuity of the inverse

If a function $f$ with domain interval is $1-1$ and continuous at an inner point a of the interval its inverse is continuous at $f(a)$?
0
votes
0answers
43 views

Solve a matrix product without computing the inverse

If I have these matrix relationship expressed as a factorization: $$\mathbf{A}=\mathbf{B}\cdot\mathbf{C}$$ where they are $\mathbf{A}\in\{0, 1\}^{m , n}$, $\mathbf{B}\in\{0, 1\}^{m , r}$ and $\...
3
votes
5answers
40 views

One-to-one function's inverse

I've been trying to solve this question for a while and couldn't find the correct way. We're looking for the inverse of the given function $r$ in terms of $f^{-1}$, where $r$ is defined by: $$r(x) = 1 ...
2
votes
0answers
26 views

Inverse Fourier transform using laplace

We have to solve the inverse FT of $$\frac{1}{1+4w^2}$$ I tried to do the synthesis but got mediocre results. However this term screams laplace to me. I can see a sine in there. The last lecture they ...
0
votes
4answers
38 views

How to verify the inverse of a polynomial in mod polynomial?

This is in $F_2$. This might sound silly but I know that the inverse of $(x^3+x)$ in mod $(x^4+x+1)$ is $(x^3 + x^2)$ but I am not sure how to verify that. It should be that when I multiply $(x^3+x)$ ...
0
votes
1answer
18 views

Mathematical function with input not in definition

I have just come across a definition for a mathematical function where in the input is not part of the function definition. This is a simplified variation of the function: $f(x) = \sin(a) + b$ ...
1
vote
2answers
63 views

Is it possible to inverse a sum of exponents

I have a problem, I need to inverse a sum of exponents. Is it possible? I have this function $y = 0.84826731\times e^{-1.10973369x} + 0.17939312\times e^{-0.1902204x} + 0.02965983\times e^{-0....
0
votes
0answers
17 views

Series Reversion for $n$ power series

I have $n$ functions with power series representation as $F_i(X)=\sum_{k_1,\dots k_m}a^{i}_{k_1,k_2,\dots k_m}x_1^{k_1}x_2^{k_2}\dots x_n^{k_n}$, where $X=[x_1,x_2,\dots,x_n]$ and $F(x)=[F_1(X),\dots,...
0
votes
0answers
16 views

How can I calculate the inverse fourier transform of $jw$

I am trying to solve $h_I(t)$, which is satisfying $h(t)*h_I(t)=\delta(t)$. e.g. If $h(t) = \delta(t+c)$, then $h_I(t)=\delta(t-c)$. If $h(t) = u(t)$, then $h_I(t)=\delta'(t)$. Q) What is the ...
0
votes
1answer
23 views

How do I Inverse Laplace $\frac{(s+1)^3}{s^4}$

I missed a class this week in maths and been a bit lost since with Inverse Laplace, how do I go about finding the Inverse laplace of: $$\frac{(s+1)^3}{s^4}$$ Do I simply expand the numerator? then ...
0
votes
0answers
30 views

Use conformal mapping to get a heart shape from a square

I use what I call "inverse" conformal mapping in order to properly handle integer locations of output pixels. In other words, if z = x + iy = (x, y) is an output pixel, then I use f(z) = (X + iY) ...
0
votes
1answer
46 views

Polynomial modular inverse

I am trying to understand the modular inverse of a polynomial. Let $A , Q$ be polynomials; what is the polynomial $B$ such that $A B = 1 \pmod Q$? I tried searching articles from Finding inverse of ...
1
vote
2answers
70 views

Given that $ A$ is a square matrix such that $ A^2 -4A -3I =0$ how do I find $(A+2I)^{-1}$

The way I tried to solve this is to find out how much $(A+2I)$ equals then find it's inverse so: $$A^2 -4A -3I == A^2 -3A-A -2I-I=0$$ $$A^2 -3A-I = (A+2I)$$ Do I simply just inverte the left ...
0
votes
1answer
21 views

matrix algebra with invertibles

Im asked to solve for X given the equation $$ (A^{-1}X)^{-1} = (AB^{-1})^{-1}(AB^2) $$ What I have done so far is: $$ ((A^{-1}X)^{-1})^{-1} = ((AB^{-1})^{-1}(AB^2))^{-1} \\A^{-1}X = ((AB^{-1})^{-1}(AB^...
2
votes
4answers
47 views

Inverse function of $f(x) = \frac{x+5}{x-2}$

Find the inverse of the function $f(x) = \frac{x+5}{x-2}$ Here's what I have so far: $y = \frac{x+5}{x-2}$ $x = \frac{y+5}{y-2}$ $(x)(y-2) = (y+5)$ but this seems to be a dead end. How should I ...
1
vote
0answers
29 views

inverse square root of band matrix

this is my first post in this web site and I hope that I find an answer to my question. I am trying to find a closed-form expression for the inverse square root of the following symmetric band matrix ...
3
votes
3answers
139 views

What is the inverse function of $e^x +x$?

As the natural $\log(x)$ function is the inverse of the exponential $e^x$ and $\log(x +1)$ is the inverse of $e^x - 1$, what it the inverse of $e^x + x$?
0
votes
2answers
123 views

what does it mean for the transpose of a matrix to be the negative of the matrix?

Say I have matrix A, if the transpose of A is equal to A then A is symmetrical. but what does it mean if the transpose of A = -A and can I know something about such a matrix inverse?
0
votes
1answer
29 views

I'd like to know the inverse of this function:$ y = 100x + 5 (x - 1) (x / 2)$

I'm developing a game with a ranking system and I'm using the formula $$y = 100x + 5 (x - 1) (x / 2)$$ to figure out how much XP is needed to obtain a certain rank. Now the problem is, I also need to ...
0
votes
0answers
10 views

inverse and unitary invariance

I want to simplify the expression $(\mathbf{V}^H \mathbf{X} \mathbf{V} )^{-1}$, where $V \in \mathbb{C}^{M\times d}, M>d$ is unitary but not square, i.e., $\mathbf{V}^H\mathbf{V}=\mathbf{I}_d$ but $...
0
votes
0answers
43 views

Consider the relation … What are the domain and range of R? Define the inverse relation. What are its domain and range?

Consider the relation R={(x,y)∈ℝ×ℝ:y=2x}. What are the domain and range of R? Define the inverse relation. What are its domain and range? So I was thinking since there is no set given that it would ...
2
votes
2answers
39 views

Inversion of a Block Matrix

Let $S$ to be a symmetric and positive semi-definite matrix of size $n$. What is the inverse of the following block matrix $$ M_{2n\times 2n}= \begin{bmatrix} aI+S & -I\\ -I & aI+S \end{...
1
vote
2answers
95 views

How to find the range of $1 / (1+x^2)^{1/2}$?

How to find the range of $$\frac{1}{\sqrt{1+x^2}}$$? Ok. I've revised the (easy theory). I would like to complete the exercise finding the derivative of f(x) and setting equal to zero. I do it ...
0
votes
0answers
21 views

Special structured matrices manipulation and inverse

As apart of a bigger analysis I'm doing, I obtained symmetric matrices of a special structure such as $$ \mathbf{A}=\left[\begin{array}{cccc} \alpha_{2}G_{11} & \alpha_{1}G_{12} & \cdots & ...
3
votes
0answers
30 views

On generalised inverse

Let $A$ be a positive matrix, may not be invertible. I define its generalised inverse as \begin{equation} A^- = \lim_{n\rightarrow \infty} \left( \frac{1}{n} I + A\right)^{-1}. \end{equation} Lets ...
1
vote
1answer
47 views

How to simplify expressions like $\sinh(4\,\text{arcsinh}(x))$?

I understand that expressions like $\sinh(\text{arcsinh}(x))$ simplify immediately and expressions like $\sinh(\text{arccosh}(x))$ tend to simplify after some algebra. However I cannot work out how ...
0
votes
0answers
101 views

Find the Inverse Matrix of a Transformation

Let $ f: \mathbb R^3 \rightarrow \mathbb R^3$ be a linear mapping which reflects $\bar{x}$ over the plane $x_1+x_2+x_3 = 0$ . You are given the standard matrix for $f$ is: $$ \frac{1}{3}\begin{...