Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
3answers
39 views

Invertibilty of the inverse matrix

Suppose I have a matrix $A$ which is invertible. Call $A^{-1}$ the inverse of $A$. Is $A^{-1}$ invertible just because is the inverse of another matrix? Suppose also I know the determinant of $A$, $...
-2
votes
2answers
63 views

Solve for x in $ x^2 + y^2 = 1 $ and $ x \pm y = \frac \pi4 $

Solve for x in $ x^2+ y^2 = 1 $ and $ x \pm y = \frac \pi4 $ I tried solving this by substitute method. And using the quadratic formula, but that create lots of cases. The original problem was to ...
1
vote
2answers
45 views

prove that $ 2 \arctan({\csc \arctan x - \tan \text{arccot }x}) = \arctan x $

Prove that $ 2 \arctan({\csc (\arctan x) -\tan (\text{arccot }x)}) = \arctan x $ x is not equal to zero. So, to solve this I tried I made two condition $ x \gt 0 $ and $ x \lt 0 $ If $ x \gt 0 $ ...
1
vote
1answer
26 views

Moore-Penrose pseudoinverse and multiplication by diagonal matrix

Let $A \in \mathbb{R}^{n \times p}$, let $D$ be a diagonal matrix with positive entries. $\dagger$ denotes the Moore-Penrose pseudoinverse. Is it true in general that: $$(A^\top D A)^\dagger A^\top D ...
1
vote
0answers
34 views

Matrix of integer powers

Is there a name for the square matrix ($j=0...n$, $k=0...n$) $M_{jk} = j^k$ (with special case $M_{00}:=1$) and is there a closed general formula for its inverse? I have stumbled upon this while ...
5
votes
5answers
72 views

Finding an inverse function (sum of non-integer powers)

I have a function: $$f(x)=x^{2.2} + (1-x)^{2.2}$$ It is defined on the interval $[0,1]$. Minimum: $x=0.5, y=2*0.5^{2.2} = 2^{-1.2}$. I want to find an inverse for it. Since the function has two "...
0
votes
1answer
29 views

Inverse of symmetric matrix that is almost diagonal

I have an $N\times N$ matrix $\mathbb X$ with entries: $$X_{ij} = x_i\delta_{ij} + y_i(\delta_{i+m,j}+\delta_{i,j+m})$$ where $1 \le m \le N$, and $x_i,y_i$ are given numbers. Is there an ...
3
votes
1answer
69 views

Inverse of a symmetric tridiagonal matrix

I have a symmetric $n\times n$ matrix $\mathbb A$ with entries: $$A_{ij} = (a_i + a_{i-1})\delta_{ij} - a_i\delta_{i,j-1}-a_{j}\delta_{i-1,j}$$ where $a_0,\dots,a_n$ are given positive numbers. Is ...
1
vote
2answers
47 views

Are all right inverses of a given function, injective?

Suppose we have a function P, I want to show that all right-inverses of P are injective. I know that if a function has a left-inverse, then it bound to be injective. Isn't it correct that all right-...
0
votes
2answers
45 views

Number of solution of the equation $\cot^{-1}{\sqrt{4-x^2}+ \cos^{-1}{(x^2-5)}}=3π/2$

Number of solution of the equation $ \cot^{-1}{\sqrt{4-x^2} + \cos^{-1}{(x^2-5)}}=3\pi/2$ $$ \cot^{-1}{\sqrt{4-x^2}+ \cos^{-1}{(x^2-5)}}={3π/2}$$ Taking sine both side and solving this is I get $$1 +\...
3
votes
0answers
37 views

Deriving a multivariate inverse function

In my math assignment I have to find the inverse of $$ f(x_1, x_2) = \left(\ln \left(\frac{x_2}{x_1}\right), x_1^2 + x_2^2\right) $$ Now I already have looked into this, and came up with the ...
1
vote
1answer
45 views

(Tridiagonal) Inverse of a matrix

Given this $n \times n$ matrix: $$ A= \left(\begin{matrix}a_1&a_1&...&a_1\\a_1&a_2&...&a_2\\\vdots& &\ddots &\vdots\\a_1&a_2&...&a_n\end{matrix}\right) $...
3
votes
5answers
129 views

Show that $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$.

Show that $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$. Once you have found the inverse like here, the verification is trivial. But how do you come up with such an inverse. Do I just try with general ...
2
votes
1answer
65 views

Definite integral containing 2 trig functions and a square root function

$$\int_{-\pi/4}^{\pi/4}\bigl(\cos x+ \sqrt {1+x^2}\sin^3x \cos^3x \bigr)\, dx $$ This question is from a math GRE practice test I've tried to solve this integral for 2 days... starting to think it ...
0
votes
1answer
63 views

Finding analytical solution for Matrix inverse with an unknown

I need an analytical solution for finding inverse of a matrix in terms of an unknown. I will use following example to pose my question EXAMPLE: $\mathbf{H}$ is a $49\times 49$ matrix. $rank(\...
2
votes
0answers
45 views

How to compute the inverse of $I + A$, where $A$ is skew-symmetric?

As we all know that if $A$ is Real skew Symmetric Matrix, then $I-A$ is Non singular. But if we really want to compute that inverse, I wonder if there is a closed form solution.
3
votes
1answer
76 views

Invertibility of a matrix in portfolio optimization

Let $A$ be an $n\times n$ symmetric matrix with non-negative entries. Let $\mathbf{1}$ be the column vector of dimension $n$ with all entries being $1$. Consider the $(n+1)\times (n+1)$ matrix $$ B= \...
-1
votes
2answers
35 views

Why if $B$ isn' t invertible then there exists $x\neq 0$ such that $Bx=0$? [closed]

If $B$ isn' t invertible then there exists $x\neq 0$ such that $Bx=0$. So $B$ would be $0$? Why?
0
votes
3answers
79 views

Inverse of $α∈Z_3(α)$ where $α^3+α^2+2=0$ [on hold]

Find the inverse of the element in the given field. The field is a finite extension F(α). Express your answer in the form $a_0 + a_1 α + \cdots + a_{n−1} α^{n−1}$, where $a_i ∈ F$ and $[F(α):F]=n$. $...
1
vote
1answer
26 views

How does this matrix math with complex numbers work/where is the mistake.

I am a beginner to these types of math, so you will have to forgive me if this question has an obvious answer. I just learned how to get the inverse of a matrix, so I thought it would be interesting ...
0
votes
2answers
42 views

Inverse of $2 − 3i \text{ in } \Bbb Q(i).$

Find the inverse of the element in the given field. The field is a finite extension F(α). Express your answer in the form $a_0 +a_1α+ ···+a_{n−1}α^{n−1}$, where $a_i$ ∈ F and [F(α):F]=n. $$2 − 3i \...
0
votes
0answers
26 views

MATLAB/EXCEL Inverse Cumulative Distribution Function (ICDF) for Negative Binomial Distribution

I have some old MATLAB code that uses the ICDF() function call (inverse cumulative distribution function). I need to translate the following into an equivalent ...
0
votes
1answer
43 views

Writing $B^{-1}$ in terms of $I, B, B^2$. [closed]

Give an expression for an inverse of a matrix, i.e., $B^{-1}$, using only $I, B, B^2$. This seems like it shouldn't be too difficult but I am unable to find anything close.
1
vote
2answers
35 views

Proving $\tan^{-1}\frac{1}{2\cdot1^{2}}+\tan^{-1}\frac{1}{2\cdot2^{2}}+\cdots+\tan^{-1}\frac{1}{2\cdot n^{2}}=\frac{\pi}{4}-\tan^{-1}\frac{1}{2n+1}$

By using Mathematical Induction, prove the following equation for all positive integers $n$: $$\tan^{-1}\frac{1}{2 \cdot 1^{2}} + \tan^{-1}\frac{1}{2 \cdot 2^{2}} +\cdots+\tan^{-1}\frac{1}{2 \cdot n^...
-1
votes
0answers
31 views

Inverse function to the exponential integral

$Ei(-ln(f(x)) = c + ln(ln(x))$ where $Ei(x)= $ - $\int_{-x}^{\infty} \frac{e^{-t}}{t} dt $, where $x $ be a non zero real (called exponential integral ). Is there a way to express $f(x)$? If yes, ...
0
votes
0answers
18 views

Finding the inverse of function for $x>0$

I have the following function $$f(x) = \frac{1}{2x}\left(\sigma(x) - \frac{1}{2}\right)$$ where $\sigma(x)$ is the sigmoid function $$\sigma(x) = \frac{e^x}{1 + e^x} = \frac{1}{1+e^{-x}}$$ I ...
1
vote
2answers
70 views

Given to find $\arccos\left(\cos(\frac{14 \pi}{3})\right)$

Okay so $\arccos\left(\cos\left(\dfrac{14 \pi}{3}\right)\right)$ can be written as $\arccos\left(\cos\left(4 \pi+\dfrac{2 \pi}{3}\right)\right)$ yielding answer as $\frac{2\pi}{3}$ But why can't we ...
1
vote
2answers
47 views

How to prove that in an abelian group $-(-a) = a$?

I have to prove that in an abelian group $-(-a) = a$, and the only hint given is that the inverses are unique. My attempt is as follows: $-(-a) = a$ is equivalent to $-(-a) - a = 0$, but I don't ...
4
votes
4answers
59 views

Let $A$ be an $n*n$ matrix such that $A^3=A^2+A-I$. If $A$ Is diagonalizable Show that $A=A^{-1}$

Let $A$ be an $n*n$ matrix such that $A^3=A^2+A-I$. Show that $A$ is invertible Suppose in $A$ is diagonalizable. Show that $A=A^{-1}$ For the first part I managed to do it by a ...
0
votes
1answer
37 views

Inverse of an Elementary Matrix

Assume we have a 3x3 matrix like: A = 9 8 7 6 5 4 3 2 1 We are applying an Elementary matrix E to A: ...
0
votes
0answers
12 views

Calculating a matrix-vector product efficiently without inverse operation

Given a matrix $A=I+K^{-1}BB^TK^{-1}$ where K is a discrete Laplacian, and B is a sparse uniformly distributed random matrix, how can I calculate matrix-vector products $Av$ efficiently without any ...
0
votes
0answers
9 views

2D convolution with Gaussian using Fourier transform

I was solving 2D diffusion equation with initial condition \chi(x,0)=1 in the circle centered at origin with radius r. Equation I want to solve To solve this equation efficiently, I need to use ...
2
votes
1answer
36 views

Compute inverse of ill-conditioned matrix

I need to compute inverse of a matrix that is highly ill conditioned and nearly singular. I tried using Jacobi preconditioning, a method to add a scalar value to the diagonal entries of the original ...
0
votes
0answers
8 views

Finding inverse operator (Green function)

I'm working on some quantum field theory and have to operate on a field with the following operator: $$ (x^\mu \partial_\mu + 1)^{-1} $$ I've been trying to find an explicit form of this operator, ...
0
votes
1answer
37 views

Guaranteed invertible matrix

Let it be two $m \times n$ matrices: $A$ and $B$, where $m,n \geq2$. Rows of these matrices are linearly independent. So, which matrix is guaranteed invertible: $AA^T$, $B^TB$, $AB^T$, $A^TB$? I ...
0
votes
0answers
13 views

Pseudo inverse of a singular matrix without any linearly independent rows or columns

It is given here that "when A has linearly independent columns (and thus matrix $A^{*}A$ is invertible), $A^{+}$ can be computed as: $A^{+}=(A^{*}A)^{-1}A^{*}$ " and there is a similar expression for ...
-3
votes
1answer
28 views

Linear Algebra Matrix Skew Symmetric [closed]

Recall that an nxn matrix is called skew-symmetric if A^T=-A. a)Prove that for all x that are in R^n we have x^TAx=0 (note: x^T Ax is a scalar for any nxn matrix A) b) Prove that I+A is invertible My ...
0
votes
1answer
33 views

Linear Algebra Inverse [duplicate]

Let $𝐴$ be an $𝑛×𝑛$ matrix and $𝑂$ be the zero $𝑛×𝑛$ matrix. a) Suppose that $𝐴^2=𝑂$. Prove that $𝐼+𝐴$ is invertible. b) Suppose that $𝐴^𝑘=𝑂$ for some $𝑘$. Prove that $𝐼+𝐴$ is ...
3
votes
2answers
43 views

Given that $A+B$ is invertible, prove or disprove $A(A+B)B = B(A+B)A$ as well as show that $A(A + B)^{-1}B = B(A + B)^{-1}A$

I have tried coming up with a counterexample for the first one, but it has worked each time so my intuition is that the first statement is true. I've tried using the fact that $\det(A+B)$ does not ...
0
votes
0answers
28 views

Need help understanding an equation: composition, addition and inverse

I have found an interesting paper on a digital image registration algorithm. There are many equations in the paper that I only understand partially, but there is a particular one I would like to ...
1
vote
1answer
41 views

Inverse function of $ax + bx^3$

I am trying to find the inverse of the function $y = f(x) = ax + bx^3$, i.e. $x = f^{-1}(y)$. (The equation arises in the modeling of a certain type of transmission used in robots) Looking at the ...
1
vote
1answer
25 views

Elements reduction in a general case

If $G$ is a group, then each element has an inverse and $\forall x, y, z \in G, xy = xz \Rightarrow x^{-1} \cdot xy = x^{-1} \cdot xz \Rightarrow 1 \cdot y = 1 \cdot z \Rightarrow y = z$ However, we ...
0
votes
0answers
40 views

Invertibility of the Schur complement when $D=0$

Suppose we have a partitioned matrix $$X = \begin{bmatrix} A & B \\ C & O\end{bmatrix}$$ where $O$ is a zero matrix of proper dimensions and where $B$ and $C$ are nonsquare matrices. Also ...
-3
votes
1answer
38 views

What's inverse function of f(x,y) = (2x+3y,3x+2y)

The question is in the title - assuming that f: Q X Q -> Q X Q, what's inverse function of f(x,y) = (2x+3y,3x+2y)? For the life of me, I just can't figure it out.
0
votes
3answers
36 views

Proving a matrix identity

Let $M \in \mathbb{R}^{n\times n}$ with $\|M\| < 1$. Show $$(I - M)^{-1} = I + M(I - M)^{-1}.$$ How can I do this? I tried starting with the equality $$(I - M)(I - M)^{-1} = I, $$ Then I ...
1
vote
2answers
53 views

Group inverse of $\textbf{A}\ast\textbf{B}:=\textbf{B}\textbf{A}+\textbf{A}+\textbf{B}$

This question came up on a recent linear algebra exam of mine, and it's been bothering me ever since. The group is defined such that every element plus the identity matrix is invertible: $$(G,\ast):=\...
1
vote
1answer
40 views

Inverse Laplace of function cos(a s)

In a physical problem, I need to calculate the inverse Laplace of function cos(a s), in which a is a real non-negative value. Is there an analytical or a numerical way to calculate the inversion?
0
votes
0answers
16 views

Expression involving inverse of block matrices and matrix exponentials

I'm struggling to simplify $B$ which is given by $$B=\left(A^{-1}\right)^TS\left(A^{-1}\right)$$ with S a symmetric matrix of size $2m \times 2m$ and A a matrix given by $$A=\left[\begin{matrix} Ve^{\...
1
vote
1answer
50 views

How to show that the following 2 matrices are conjugate?

How to show that the following 2 matrices are conjugate? \begin{bmatrix} z & 0 \\ 0 & z^{-1} \end{bmatrix} And \begin{bmatrix} z^{-1}& 0 \\ 0 & z \end{bmatrix} I know the ...
0
votes
3answers
29 views

diagonal matrix and invertible matrix proof [closed]

I am given the following proof question: Let $A \in {\mathbb R}^{n\times n} $.` Show that there exist invertible matrices $B$, $C$ such that $A=B+C$. I believe it has something to do with ...