# 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. ...
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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 ...
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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?
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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 ...
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### 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 ...
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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 ...
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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 $i,j$ ...
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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 ...
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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 $p\times p$ $M'$...
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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 ...
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### 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 $\iff$ ...
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### 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 ...
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### 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{...
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### 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 ...
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 ...