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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Interesting Simple Integral Over the Unit Interval of the Inverse Regularized Incomplete Gamma Function. Non-Integral Form Needed. Closed Form?

I have recently used, $\Bbb {here}$, with the Regularized Incomplete Gamma Function. This then made me wonder about its inverse. This function can easily be integrated with respect to its second ...
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Given two bases vectors for $\mathbb C_3$ find change of bases matrix from $e_1,e_2,e_3$ to $a_1,a_2,a_3$.

Given two bases vectors for $\mathbb C_3$ find change of bases matrix from $e_1,e_2,e_3$ to $a_1,a_2,a_3$. What we did in class was to stack $e_1, e_2, e_3$ into columns of $A$ and $a_1,a_2,a_3$ into ...
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$A$ orthogonal matrix with the eigenvalue not equal to -1. Proving A is expressible as $(I+S)(I-S)^{-1}$ where $S$ is skew-symmetric matrix. [closed]

If $A$ be an orthogonal matrix with the eigenvalue not equal to $1$. Prove that $A$ is expressible as $(I+S)(I-S)^{-1}$ where $S$ is skew-symmetric matrix.
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Shifting a positive semi-definite matrix

Let $n\in \mathbb{N}$, and $A\in \mathbb{R}^{n\times n}$ be a semi-positive definite matrix. What can we say about the matrix $$ A_h:= A+(h-1)I, $$ where $I$ is the identity matrix, and $h>0$ (...
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39 views

Inverse matrix calculation/transformation

When does the following hold? $$\frac{v^{\top} A v}{v^{\top} v} = \frac{v^{\top} v}{v^{\top} A^{-1} v}$$
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Concavity of matrix inverse function

I am trying to prove the following result: f(X) = $1/(\iota'X^{-1}\iota)$ is a concave function for X a positive definite (p.d.) $n \times n$ matrix, and $\iota$ denotes the vector of ones of length $...
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2answers
59 views

Proving invertibility of $A^{T}A$

Take the matrix $A$, which is $m\times n$ and of rank $n$. Hence, a full column-rank matrix. I need to show that $N(A^{T}A) = N(A)$, and deduce that $A^{T}A$ is invertible. Since A is a full column-...
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Left inverse of a matrix and a full column rank

Dr Strang in his book linear algebra and it's applications, pg 108 says ,when talking about the left inverse of a matrix( $m$ by $n$) UNIQUENESS: For a full column rank $r=n . A x=b$ has at most one ...
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Mistake when calculating modular inverse using Euclid's algorithm

So I've attempted calculating the modular inverse of $3$ modulo $68238256$, but my answer is wrong. I know the answer should be $45492171$, but I keep getting $22746085$. I can see that $68238256 - ...
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How to extract B from ABA^T where A is not square

So I have this value in the form of ABA^T where A is 3 by 2 and B is 2 by 2. I want to retrieve B, but since A is not square, it does not have an inverse. Is there a way to do it?
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Getting an equation involving logarithm into explicit form

I am at the final part of a problem where I have derived $t+c=\frac{1}{\sqrt2}\log\left(\frac{x}{2+\sqrt{4-2x^2}}\right)$ where $c$ is a constant, and now I need to express it in explicit form $x(t)$. ...
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1answer
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How to prove a Generalized Inverse Matrix (g-inverse) using rank [closed]

Let $A_{m \times n}$ matrix , rank($A$) = $r$ . Let $B$ and $K$ non singular matrices (order $m$ and $n$ respectively) such that: $$ A=B \begin{bmatrix} Ir & 0 \\ 0 & 0 \end{...
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Inverse of Definite Integrals?

We know that the inverse of an integral is the derivative, but what happens if the integral is definite (meaning: it has bounds/limits from-to)? For example we have the function $g(x) = \int f(x) dx$, ...
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Inverse matrix of an element of the span of matrix powers

I stucked at the point b) of this problem: Let $A$ be a square matrix of dimension $n$. Let $M(A) = \operatorname{span}\{A^i\mid i\ge0\}$ be a subspace of the vector space of all matrices of degree $n$...
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Zeros of $f:R^2 \rightarrow R$ with Newton-Raphson?

I would like to find zeros of $f:R^2\rightarrow R$ applying the Newton-Raphson method but I got stuck in solving the linear approximation equation for $\textbf{x}$. Let $\textbf{x}\equiv(x,y)$ and $\...
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Inverse matrix of an $n$x$n$ matrix, which is given below.

Now, I'm stuck on how to get the inverse of this matrix. I tried $3$x$3$ version on this matrix but I still have difficulty on finding a pattern on getting the inverse matrix. $$V= \displaystyle{\...
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Inverse matrix of a 2n x 2n matrix given below.

I have trouble on finding the inverse matrix of $V$, given below. I tried finding first the inverse matrix to the case where $n=2,3$. But, I cant find a pattern that will lead me to the general one. ...
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Determining the inverse of the Baker's map

The Wikipedia article for the Baker's map says that the map is invertible. The map is given by $$ (x', y')= \left(2x-\left\lfloor 2x\right\rfloor \,,\,\frac{y+\left\lfloor 2x\right\rfloor }{2}\right). ...
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Relation between a scalar function and its “inverse” gradient

I have a surface defined by the implicit formula: $$ F(\boldsymbol{x}) = 0 $$ where $ \boldsymbol{x} \in \mathbb{R}^n$ and $F : \mathbb{R}^n \to \mathbb{R} $ (actually I have $n=3$, but if the result ...
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1answer
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Can inverse mod answers be negative or must they be in a “range”?

Rebound from this question: Debug back-substitution in extended Euclidean algorithm My professor is telling me the correct answer to modular inverse of $28$ mod $45$ is $37$ and NOT $-8$. He says it ...
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Matlab Problem: Badly scaled matrix, very small condition number RCOND when using Chebyshev discretization

I am using MATLAB for the following problem. I have the following problem statement: LHS * q = RHS * f. This can be rewritten to q = H *f with H = LHS\RHS; Hereby q and f are vectors, LHS and RHS ...
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Solving matrix equations for x

I have a bit of a problem with some matrix equations: Task 1: $X,A$ arbitrary matrices, $A$ invertible. Solve for $X$: $XA+A^T=I \Longleftrightarrow XA = I-A^T \Longleftrightarrow X = IA^{-1}-A^TA^{-1}...
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2answers
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Matrix Inverse of Outer Product of a vector with itself

Suppose I have a vector $x \in \mathbb{R}^n$ and the matrix $$ A = \lambda x x^\top \qquad\lambda \in \mathbb{R}\backslash\{0\} $$ I would like to find an expression for its inverse. Is there a simple ...
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bounded spectral norm of $\left(\frac{1}{n} A A^{H}-z I_{m \times m}\right)^{-1}$ for any complex-valued 𝒛 with a nonzero imaginary part.

Let $\boldsymbol{A} \in \mathbb{C}^{m \times n}, m \geq n, \operatorname{rank}\{\boldsymbol{A}\}=n$ I want to Show that for all sizes the matrix $\left(\frac{1}{n} A A^{H}-z I_{m \times m}\right)^{-1}$...
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1answer
40 views

$A = B+ C$ then $(B^{-1} - A^{-1})$ and $(C^{-1} - A^{-1})$ are positive semidefinite

Let $A$, $B$ and $C$ invertible and symmetric square real matrices of dimension $n$. I want to show that if $A = B+ C$ then $(B^{-1} - A^{-1})$ and $(C^{-1} - A^{-1})$ are positive semidefinite. For a ...
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25 views

Confusion about finding eigenvectors from matrix multiplication

I'm doing the following question here: Find the determinant of $A$=\begin{bmatrix} 6 & 2 & 2 & 2 &2 \\ 2 & 6 & 2 & 2 & 2 \\ 2 & 2 & 6 & 2 & ...
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37 views

Uniform Continuity and Inverse

Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous. Prove the following statements: $(a)$ For any $\epsilon>0$, there exists a piecewise constant function $s:[a, b] \rightarrow \mathbb{R}$ such ...
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Taylor Series Corollary

The property I am about to describe seems to me to naturally derived from the definition of Taylor series : Let $ g $ and $ f $ be two real analytic functions such that $ g (x): = f ^ {- 1} (x) $ $$g(...
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1answer
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If $A$ is non-singular, prove $|A| = \frac{1}{|A^{-1}|}$? [duplicate]

I am unsure on how to prove the following problem: If a $2 \times 2$ matrix $A$ is non-singular, prove $|A| = \frac{1}{|A^{-1}|}$ I know that $|A|\cdot|A^{-1}| = I$ but i’m not sure where to go from ...
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Approximate matrix inverse by Fourier approach

Given a hermititan matrix $A$ with the possibility to generate $e^{-iAt}$ for $t\geq 0$ how would I proceed to approxiamte: $A^{-1}\approx\sum_j\alpha_je^{-iAt_j}$ $\quad$ with $\alpha_j\in\mathbb{C}$...
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When does one invertibility condition suffices?

It is often the case that, to prove $f$ being the inverse morphism of $g$, one has only to show $fg = id$ and the other direction ($gf = id$) is guaranteed to be true -- e. g. when considering vector ...
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How to find the inverse of a linear combination of unitary matrices?

Given scalars $\alpha_1, \alpha_2, \dots, \alpha_\ell \neq 0$ and unitary matrices $U_1, U_2, \dots, U_\ell$, let $$ A := \sum_{k=1}^{\ell} \alpha_k U_k $$ where the $\ell$ coefficients $\alpha_1, \...
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Show that $A \setminus f^{-1} (C) = f^{−1}(B \setminus C)$ when $f : A → B$ is a map with $C ⊆ B$ .

Show that $A \setminus f^{-1} (C) = f^{−1}(B \setminus C)$ when $f : A → B$ is a map with $C ⊆ B$. Here is what I have been up to: $x \in$ of $A \setminus f^{-1} (C)$ $x \in$ of $A \cap [f^{-1} (C)]^{-...
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Is the inverse of a restricted compact operator unbounded?

Suppose we have two separable Hilbert spaces $\mathbb{H}_{1},\mathbb{H}_{2}$ and the compact operator $\mathscr{T}:\mathbb{H}_{1}\to\mathbb{H}_{2}$. We know that since $\mathscr{T}$ is compact, its ...
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Inverting a Block-Toeplitz matrix with the Sherman-Morrison formula

Suppose we are given the following Block-Toeplitz matrix: \begin{eqnarray} T=\left(\begin{matrix} A & 0 & ... & 0\\ B & A & ... & \vdots\\ \vdots & \ddots & \ddots &...
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What is inverse Fourier transform of e^-|α|?

I am using the mathematical convention but my answer in any case is not matching. I have attached my approach below.
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67 views

How to find the inverse of an integral

Let the function $g$ be strictly positive (or strictly negative). Let $f\left( x\right) =\int _{a}^{x}g\left( t\right) dt$. How to find $f^{-1}\left( x\right)$ in terms of $g$? Let's use $f\left( x\...
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Laurent expansion of inverse of quadratic matrix polynomial

Let $A$ and $B$ be two square matrices of the same size where $A$ is invertible and the kernel of $B$ is $1$-dimensional. Consider the function $f:\mathbb C \to \mathbb M$ defined by $$f(z) = z^2I + ...
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Dimensions of identity and zero matrices (in brackets together) in these equations?

This is from page 104 of this dissertation: Virtual Analog Modeling of Audio Circuitry Using Wave Digital Filters, Kurt Werner It's been a while since I've done matrix algebra. I'm interested in I ...
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General inversion for complex block Toeplitz matrices

I have been looking at inversion methods of block Toeplitz matrices, and found the paper by Akaike for real block Toeplitz matrices. Is there any good reference to look at inversion of complex-values, ...
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Inverse of continuous matrix-valued function

Suppose that $A\colon \mathbb{R}\to \mathbb{R}^{n\times n}$ is a matrix-valued funciton, and is always positive definite on the domain. Also let $A(x)$ be continuous on $\mathbb{R}$ (i.e., every ...
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In a group if $ab=e$ then is it always true that $ba=e$? [duplicate]

I think it's not true. But since in a group, inverse for each element exist then, we can write: $ab=e$ Multiplying by $a^{-1}$ on both sides $b=a^{-1}e=a^{-1}$ This implies that $ba=e$ Am I missing ...
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subspace topology and universal property

We let $f$ be a continuous function $f:X\rightarrow Y$ where $X$, $Y$ are two topological spaces. If $g:T\rightarrow Y$ is a continuous function s.t for every $t\in T$ there exists $x\in X$ s.t $g(t)=...
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Inverse of an invertible upper triangular matrix is upper triangular

Let $A$ be a ring and $B:=(\lambda_{ij})_{ij}$ an upper triangular $n\times n$ matrix over $A$ whose diagonal elements are invertible. I want to show that $B^{-1}$ exists and is upper triangular. Let $...
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81 views

If $A^3 = 2I$, prove that matrix $A - 2^{\frac{1}{3}}I$ is not invertible

If $A$ is a square real matrix and $A^3 = 2I$, how can I prove that matrix $A - 2^{\frac{1}{3}}I$ is not invertible? I know it can be solved using the characteristic polynomial of matrix $A$, but I ...
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1answer
16 views

Matrix that transforms the vector components inverse of the matrix that transforms the basis vectors

I'm starting to learn about tensors and my textbook (Riley, Hobson and Bence, 3rd) states that the matrix that transforms the vector components must be the inverse of the matrix that transforms the ...
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1answer
36 views

How to quickly inverse a permutation by using PyTorch? [closed]

I am confused on how to quickly restore an array shuffled by a permutation. ...
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1answer
117 views

Is it possible to find the inverse to this function?

Sorry if this is not a good question, but I normally don't venture to the math side of things, at least not that far where I can't stand anymore, so please forgive me if this question is not well ...
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1answer
23 views

Additive Inverse and integer modulo

I am not completely sure how inverses work with sets of integer modulo. I have just started to learn about them. I have tried some practice problems, though I am not sure if my approach is correct in ...
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17 views

A Sherman-Morrison formula for mixed tensor products

There is a direct analog to the Sherman-Morrison formula for this type of rank two update to a fourth order tensor: $$ \left(C_{ijkl} + A_{ij} B_{kl} \right)^{-1} = C_{ijkl}^{-1} - \frac{C_{ijmn}^{-1}...

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