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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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Closed form formula for $(\mathbb I+R)^{-1}$ where $R$ is an orthogonal matrix?

Is there a closed form formula for $(\mathbb I+R)^{-1}$, where $R$ is an orthogonal matrix? There are formulas for inverse of matrix sums, but I couldn't find one that evaluated to a simple result. ...
AccidentalTaylorExpansion's user avatar
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Inverse Fourier Transform - convolution of exponential and rectangular window

I'm trying to get the response in the time domain of the convolution between the exponential $u(t)e^{-at}$ and the rectangular window ($u(t+1)-u(t-1)$). I had already obtained its result by ...
nickalicas's user avatar
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$P=S^{-1}$ and $Q = S[1:k, 1:k]^{-1}$. Can we write $Q$ in terms of $P$?

Let $S$ be an $n\times n$ positive definite matrix. For $k < n$, define $$P= S^{-1}\quad\text{and}\quad Q=S[1\text{:}k,1\text{:}k]^{-1}$$ where $S[1\text{:}k,1\text{:}k]$ is the principal leading ...
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Inverse formula [closed]

I have the following equation: $$y = 2^{-10}(2^{10x}+x)$$ What would be the inverse of this equation?
Roel's user avatar
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Solving for a Specific Variable in an Underdetermined System of Linear Equations

I have an underdetermined system of linear equations of the form $Ax = b$, where: $$ A = \begin{bmatrix} a_{20} & a_{10} & a_{00} & 0 & 0 \\ 0 & a_{21} & a_{11} & a_{01} &...
Omid Abasi's user avatar
2 votes
2 answers
134 views

Is $\text{Id}-T$ always invertible when $\lim_{n\to\infty} T^n = 0$?

Let $T$ be a linear map on a normed vector space over the real numbers. I know that \begin{align*} \lim_{n\to\infty} T^n = 0, \end{align*} i.e. applying $T$ "infinitely many times" results ...
Greaseddog's user avatar
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2 answers
70 views

Prove that the given function is invertible

This question is linked with this question I have $$f(x,y,z)=(x+y+z^2,x-y+z,2x+y-z)$$ and I need to prove that it is invertible at the point $(1,-1,2)$. I have read that proving such a quality ...
Superunknown's user avatar
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Invertibility of a measurable mapping from lower and upperbounds on the induced pushforward measure

Let $\Omega \subseteq \mathbb{R}$ be open and consider the standard Borel space $(\Omega, \mathcal{B}(\Omega), \mu)$, where $\mu$ denotes the Lebeasgue measure. Let $f: \Omega \to \Omega$ be a ...
Saleh's user avatar
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Numerical Inverse of an integral

Let $f: \mathbb{R} \to (0, \infty)$ be a continuous function. Define $F: \mathbb{R} \to (0, \infty)$ by: \begin{equation*} F(x) = \int_{-\infty}^x f(t)dt \end{equation*} Clearly, $F$ is strictly ...
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Iteratively finding matrix inverse from a given inverted matrix.

Let's say I need to find inverse of a bunch of matrices $A_0, A_1, ... A_n$. As the matrices are large (in my use-case), I am iteratively finding the inverse of each matrix $A_i$ using Newton matrix ...
Satya Prakash Dash's user avatar
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If $X\sim\mathcal N(x,\Sigma)$, what is $\operatorname E\left[\left\|\Sigma^{-1}(X-x)\right\|^2\right]$?

The question is in the title. If $\Sigma=\sigma^2I_d$, then we easily calculate $$\operatorname E\left[\left\|\Sigma^{-1}(X-x)\right\|^2\right]=\frac1{\sigma^2}\tag1.$$ However, the general case seems ...
0xbadf00d's user avatar
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Solving for matrix in Ax=y

What can be said about the equation $Ax=y$ where $x,y$ are known vectors? I couldn't find much information about this equation online. The problem where $A,x$ are known is trivial by simply ...
Ervin Macić's user avatar
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Is this special BLOCK upper triangular matrix diagonalizable?

Let $A$ be a block upper triangular matrix: $$A = \begin{bmatrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{bmatrix}$$ where $A_{1,1} ∈ {\mathbb{R}}^{p \times p}$, $A_{2,2} ∈ {\mathbb{R}}^{(q) \times (...
Manish Kumar's user avatar
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Change a matrix of base with respect a special base consisted of a kernel and a rowspace

I have got a matrix $B$ $m \times n$. I found a basis of its Kernel and of its rowspace so both together form a basis of $\Re^n$. I have seen in an article that you can now express a matrix $A$ $n \...
ignaciocrb's user avatar
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Proving $(S \circ R)^{-1} = R^{-1} \circ S^{-1}$ [duplicate]

I need to prove the following statement from Velleman's How to Prove It (p. 176): Suppose $R$ is a relation from $A$ to $B$ and $S$ is a relation from $B$ to $C$. Prove that $(S \circ R)^{-1} = R^{-1}...
Riccardo Iorio's user avatar
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Inverse of a Block matrix formed by two column blocks which represent orthogonal complements

I have got a matrix $B$ $m \times n$ with rank $m$. I found then the basis of its kernel and of its rowspace. Both together form a basis of $\Re^n$. With this I make the change of basis matrix $P$ ...
ignaciocrb's user avatar
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1 answer
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Inverse and Composition of Bisimulations

Exercise 63 of Rutten's The Method of Coalgebra: exercises in coinduction asks us to prove that "the collection of all bisimulation relations between two given stream systems is closed under (i) ...
msb15's user avatar
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Problem 46, section 2.5 on Introduction to Linear Algebra, 4th edition.

The question asks how does the identity $A(I+BA)=(I+AB)A$ connect the inverses of $I+BA$ and $I+AB$, and I am able to express $(I+BA)^{-1}$ as $A^{-1}(I+AB)^{-1}A$. However, I fail to see how this ...
Xiangnong Wu's user avatar
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1 answer
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Modified graph Laplacian, D + A

Consider an undirected, connected graph with positive edge weights $G$ with adjacency matrix $A$ and (diagonal) degree matrix $D$, and graph Laplacian given by $L = D - A$. $L$ is singular and is non-...
magnesium's user avatar
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1 answer
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Why is (I+A) invertible and what is its inverse? [duplicate]

Let A and B be n x n (square) matrices where A^2 = 0 and B^2 = 0. Let I be the identity matrix. (I+A) is invertible. Why? What is its inverse? Does A^2 = 0 imply that the matrix A is nilpotent, making ...
xbigf00tx's user avatar
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1 answer
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Inverse of the identity minus a block anti-diagonal matrix.

Let $A$ be an $m\times n$ matrix and $B$ be an $n\times m$ matrix. Denote by $C$ the square block matrix given by $$ C=\left[\begin{array}{cc} 0 & B\\ A & 0 \end{array}\right]. $$ I am looking ...
user_lambda's user avatar
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The inverse of a specific case of symmetric matrix (scalar product of d dimensions vectors)

The problem is the following: For $i \in [N]$, let $v_i$ be a $1 \times d$ vector and $b_i$ a scalar. Moreover, let $A$ be a $N \times N$ matrix, whose (i, j)-entry is: $$ a_{ij} = \begin{cases} \...
Funsss's user avatar
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Inverse of a matrix sum and difference in terms of known inverses

I have been working on a problem related to the inverses of matrices and would appreciate any insights or solutions. The problem is as follows: Given two invertible matrices $A$ and $B$ with known ...
triple_tactic's user avatar
1 vote
0 answers
32 views

Inverse function / mapping considering vector multiplication by matrix

Inverse function / mapping considering vector multiplication by matrix also touches symetric encryption Consider, there's a simple matrix as a mapping from R3 ➝ R3 ...
Heinrich Elsigan's user avatar
12 votes
1 answer
758 views

Taking the inverse (not the reciprocal) of both sides of an inequality

This is something I'm having a hard time finding online, but say we know that $f(x) > g(x)$ (for all inputs $x > a_{0}$ for some $a_{0}$), then would it always be true that $f^{-1}(x) < g^{-1}...
Bob Marley's user avatar
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0 answers
42 views

Verification of a demonstration

I need to know if the proof I made for the following problem is correct. Problem: If C is a matrix of order $3 \times 3$ such that $\text{rank}(C) = 2$, then $\text{det}(C) = 0$ Proof: If it must be ...
studiare's user avatar
1 vote
0 answers
89 views

How to calculate $\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$ [closed]

I wonder if there is an analytic solution for the following equation: $$\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$$ Here, $a$ is a constant. Would you please give an advice?
donggun's user avatar
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Quadratic form of a matrix where non-standard decomposition is known

Let $1\le m<<n$ for integers $m,n$. I have two symmetric matrices of size $n\times n$, say $A$ and $B$. My goal here is to know a simple form, or an upper bound on the following quadratic form: $...
2019ys's user avatar
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1 answer
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Is the product of a right-invertible, an invertible and a left-invertible matrix itself invertible?

Suppose $B \in \mathbb{R}^{(n,n)}$ is invertible and $A \in \mathbb{R}^{(n,m)}$ is left-invertible. Is $A^T B A$ invertible? I know that $A^T A$ is invertible. I've been trying to work it out using ...
knods's user avatar
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0 answers
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Is there any mathematical results stating when there are 0's in the inverse of a square matrix given 0's in the original matrix?

I am working with square invertible matrices. Denote the n-by-n matrix as $A \in \mathbb{R}^{n \times n}$. Say we know that there are some 0's in the matrix. For instance: $A_{ij} = 0$ for some $i,j$ ...
ajl123's user avatar
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0 answers
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$A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f$ is a non constant polynomial.

The Actual Question $A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f=0$ has finitely many solutions. Thoughts I understand that $f$ is a non zero polynomial, and if it's not a constant ...
Debu's user avatar
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Given $M, N$, how to find matrix $M'$ such that $M'N = NM$

Consider an arbitrary $p\times q$ matrix $N$, an arbitrary $q\times q$ matrix $M$. I do not know if $N$ is invertible - the solution is easy when it is. Is there a way to always find a $p\times p$ $M'$...
user1936752's user avatar
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1 answer
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Invertibility of an integral matrix expression

Given $A:\mathbb R^n \rightarrow \mathbb R^{n\times n}$, $x\mapsto A(x)$ invertible for all $x$. In particular, it is known that $$A(x) = \frac{\partial}{\partial x} f(x)$$ with $f:\mathbb R^n \...
Name123's user avatar
  • 49
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0 answers
16 views

Inverse Relation for Allocating an Amount

This question came up in the context of project finance. It is my original question. Quick Version A project has upfront cost $X$, and ongoing cost $Z$. (Z is less than X). There are the same $N$ ...
VISQL's user avatar
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1 answer
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Inverse function theorem generalization

In inverse function theorem, it requires $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$. Can it be applied to the case that $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ where $n>m$. In other words, how ...
William Lin's user avatar
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1 answer
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Requirements for invertibility of $A B A^T$ in constrainted dynamics

What are the requirements for matrix $A$ (that isn't a square matrix), so that the matrix $A B A^T$ is invertible, given that $B$ is non-singular? Some details for the matrices: $B$ is the $n \times n$...
MIKE PAPADAKIS's user avatar
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1 answer
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Why is $A^{-1}$ existing a necessary condition for $x$ to be unique in $Ax = b$? [duplicate]

Consider the equation $Ax = b$ where $A \in \mathbb{R}^{m \times n}$, and $x \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$. Say we know $A$ and $b$. I am wondering why, in order for us to uniquely ...
Princess Mia's user avatar
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1 vote
0 answers
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Results of invertibility of a matrix involving the Szego kernel

In the context of reproducing kernel Hilbert spaces, the Szego kernel is the function $k(z_i,z_j)=\frac{1}{1-z_j\overline{z_j}}$. Given two sets of points $\{z_1,\ldots,z_n\},\,\{w_1,\ldots,w_n\}\...
Math101's user avatar
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-2 votes
1 answer
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Find the formula for the inverse transformation of $T\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}=(x+1)^2(2c-a-b)-(3x-2)(a+b)+x(5c-b)-3c$ [closed]

I was looking to the exercise for Linear Algebra from my course materials. Stumbled upon this question, and I have no idea how to start or solve this question. As I am totally new to the realm of ...
Tanvir Ahmed's user avatar
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1 answer
48 views

Baby Rudin Theorem $4.14$ [closed]

In Baby Rudin theorem $4.14$, he says: $f(f^{-1}(E))\subset E \quad\forall E \subset Y$ and then $f^{-1}(f(E)) \supset E \quad$ if $\quad E \subset X$. I thought functions were invertible $\iff$ ...
Rudinable's user avatar
0 votes
1 answer
63 views

How to find the matrix $A$ from $C_0=A \times B$ or $C_1=B \times A$, given the singluar matrix B.

Kindly help me in the following: Let $C_0=A \times B$ and $C_1=B \times A$, where $A$ is a full rank square matrix, $B$ is a non-zero singular square matrix. Entries in $A,B,C$ are from finite ...
X.H. Yue's user avatar
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0 answers
21 views

Proof that A*M*A' is not invertible

I came across an unknown mathematical theorem in a book about the Kalman filter, but I'm not sure how to prove it. Let $M\in\mathbb{R}^{2\times2}$ be a covariance matrix (hence symmetric) and $A\in\...
Ogiad's user avatar
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0 answers
60 views

Block Matrix Inverse and Imaginary Number

I came here to ask some help regarding the following question. Let $M \in \mathbb R^{n \times n}$ be a "symmetric" and positive semidefinite matrix, and $M^{(n)}$ is obtained from $M$ by ...
jason 1's user avatar
  • 769
2 votes
1 answer
38 views

4x4 banded matrix inverse

Let $a,b,c$ be reals such that $ab-c^2\neq0$, and let $$ A = \begin{pmatrix} a & 0 & c & 0 \\ 0 & a & 0 & c \\ c & 0 & b & 0 \\ 0 & c & 0 & b \end{...
G. Gare's user avatar
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0 votes
1 answer
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What is the different between these two forms of the derivative of arcsine?

It is known that the derivative of an inverse function is given as $$ g'(y)=\frac{1}{f'(x)} \implies \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} $$ So if $\arcsin(y)$ is differentiated: $$ \arcsin(y)' = \...
thewhale's user avatar
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0 answers
30 views

A sparse matrix set where a matrix and its inverse share the identical sparse structure.

Define a sparse matrix set where a matrix and its inverse have the same sparse structure. Is such a set forming a certain group? For example, $\begin{bmatrix}a & b\\0 & c \end{bmatrix}^{-1}=\...
fibon's user avatar
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3 votes
1 answer
37 views

Writing inverse without using piecewise function [duplicate]

I am trying to find the inverse function for the given function f(x) = 3x - |x| + |x - 2|. I have already found that the inverse function can be expressed as the piecewise function f^{-1}(x) = \begin{...
Luke's user avatar
  • 86
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0 answers
49 views

Inverse tensor notation

I have been learning how to transform from one basis to another. I don't have any issues when both basis are orthogonal because I use a different formula then the one below. An issue I am having is ...
Alexander Savadelis's user avatar
-2 votes
1 answer
45 views

Why I cannot find the inverse matrix of A with the second method? [closed]

Assume A = [(1,0,-2),(-1,3,2),(5,0,-1)]is a matrix, then find the inverse of A. I have tried to find the inverse of A this way: However, I don't understand what's wrong in the following solution ...
Ignatius NG's user avatar
1 vote
0 answers
48 views

expectation on the inverse of norm-2 of complex Gaussian vector

I am working on deriving the normalization issue and face some challenges in finding the closed-form expression. Assume there is a complex vector $\mathbf{h}\in\mathcal{CN}^{P\times 1}$, where each ...
Charlie Nie's user avatar

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