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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Limit of the inverse of thedifference of two matrices in quadratic form

Let $A=(a_{ij}(x))$ and $B=(b_{ij}(x))$ be invertible symmetric positive definite ${n\times n}$ matrices, $a_{ij}\geq 0,\,b_{ij}\geq0\,\forall i,j$, also let $y$ be a $n$-vector. Note that $Q=y^T(A-...
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Proof the Laplace expansion of determinant doesn't depend on row or column of expansion

The Laplace expansion over the first row the determinant of an $(n\times n)$ Matrix $A$ is defined as $det(A)=\sum_{j=1}^{3}(-1)^{1+j}M_{1j}a_{1j}$ where $M_{ij}$ is the determinant of the $(n-1)\...
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Inverse Laplace transform of $\exp(-s^4)$ [closed]

Can anyone help me in finding the Inverse Laplace transform of $\exp(-s^4)$ Thank you
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Finding the inverse of a specific block matrix.

I have a problem where I have to calculate the inverse of a specific block matrix: $$ \begin{pmatrix}M^{-1}+B\Lambda^{-1}B^T & B\Lambda^{-1}A^T\\ A\Lambda^{-1} B^T & L^{-1} + A\Lambda^{-1}...
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Is the restriction of an inverse function invertible?

Let $f:A\rightarrow B$, where $A$ and $B$ are open sets of $\mathbb{R^n}$ for some n, be invertible. Let $C$ and $D$ be open subsets of $A$ and $B$ respectively. Is $f:C \rightarrow D$ invertible?
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How to find the inverse of elements in a field?

Normally I use the naive method: $$a^{-1} = a \cdot b \bmod p \equiv 1,$$ where b is the inverse of a. Else I love to use Fermat's little theorem: $$a^{p − 1} \equiv 1 \bmod p.$$ By multiplying both ...
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Understanding the proof for the inverse of a matrix

The following sketch of the proof is from Ron Larson's Linear Algebra book (page 129). I'm having a difficulty in seeing the "Try Verifying this". I can see that for small size matrices but not for ...
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Can $I-A$ be singular for any real $A$?

You might like this question. Is it possible that $I-A$ can be singular for any real $A$, except for $A = 0$ and $A = I$?
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Inverse of $f(x)=x^n(1-x)^k$

I am trying to find an inverse of a function \begin{align} f(x)=x^n(1-x)^k, x \in (0,1) \end{align} where $n$ and $k$ are some positive integers. I know that his function doesn't have a 'pure' ...
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37 views

Cayley-Hamilton theorem for symmetric positive definite matrix

Let $A$ be a symmetric positive definite matrix. We know that by Cayley-Hamilton theorem \begin{aligned}A^{-1}={\frac {(-1)^{n-1}}{\det A}}(A^{n-1}+c_{n-1}A^{n-2}+\cdots +c_{1}I_{n}).\end{aligned} ...
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Inverse of the difference of matrices

Let $A\in R^{n\times n}$ (invertible and symmetric), $B\in R^{k\times k}$ (invertible and symmetric), $C\in R^{n\times k}$, $D\in R^{k\times n}$, $CBD$ is (invertible and symmetric). I am trying to ...
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22 views

Limit of a inverse matrix with entries tending to zero

Let $A\in R^{n\times n}$ (symmetric), I know that, $\forall(i,j)$ $\lim_{x\to\infty} a_{ij}(x)=0$, . I want to show, if possible, that the $$\lim_{x\to\infty} A^{-1}=E,$$ where $E$ is a $n\times n$ ...
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Approximation of smooth diffeomorphisms by polynomial diffeomorphisms?

Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism? More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism. For ...
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What is the difference between left/right inverse matrix and singular value inverse matrix

As i know, both the left/right inverse matrix and singular value inverse matrix can get an inverse matrix of non square matrix. Book shows that: A matrix $A \in \mathbb{C}^{M \times N}$ if $M\geq N$...
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Reason for P subset of NP. Is it possible P > NP? [closed]

I have solved a method for solving an NP complete problem, specifically the boolean satisfiablity problem. My current optimized solution is a polynomial of degree 4 relative to B, the number of bits. ...
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Proof of matrix determinants and invereses [closed]

I am trying to figure out how to do the following proof below. Let X, and Y be two 3 by 3 matrices with real entries. If det(X * Y)=0, show that X^-1 does not exist. (This statement could either be ...
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When is a multivector in Cl(n) invertible?

My question Let $\mathrm{Cl}(n)$ be the Clifford algebra over $\mathbb{R}^n$ with the usual inner product. That is, it's the quotient of the tensor algebra over $\mathbb{R}^n$ by the ideal generated ...
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If $A$ and $B$ are $2\times 2$ matrices such that $A^2 - B^2$ is invertible, is $A-B$ invertible?

Let $A$ and $B$ be $2\times 2$ matrices such that $A^2 - B^2$ is invertible. Is $A-B$ necessarily invertible? This doesn't seem like it should be difficult but I just can't come up with a solution.
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Guaranteed invertibility of the approximated Hessian in Levenberg-Marquardt

I need to show that the approximated Hessian in the Levenberg-Marquardt algorithm is guaranteed to be invertible, whereas in the Gauss-Newton algorithm, this is not always required to be true. ...
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Is it possible that $A\cdot B= I$, but that $B\cdot A\neq I$?

I know that $A\cdot A^{-1}=A^{-1}\cdot A = I$, but is it possible for a matrix $B$ to exist such that $A\cdot B= I$, but $B\cdot A\neq I$? If that it is not the case, why not? ($I$ is the identity ...
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What is the inverse relation of R

Determine the inverse relation $R^{−1}$ for the relation $R = \{(x,y) : x + 4y \text{ is odd}\}$ defined on $\mathbb{N}$. does this mean that $R^{-1} = \{(y,x):y+4x \text{ is odd}\}$ ?
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What is the inverse of $\sqrt{ax+bx^4}$?

I am interested in the inverse function of: $$ y=\sqrt{ax+bx^4} $$ Where $a,b,x \in [0,1]$.Furthermore, $a+b=1$ but I’m not so sure that’s as relevant. This came up in the course of (physics) ...
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Relation between $\infty$-norm and 2-norm condition number of PD matrix

Consider a positive-definite matrix $A$ in $\mathrm{R}^{n\times n}$ and let $\kappa_{\infty} = \|A\|_{\infty}\|A^{-1}\|_{\infty}$ and $\kappa_2 = \|A\|_2 \|A^{-1}\|_2$, with $\|A\| _p = \sup_{x \ne 0} ...
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Why are these triangles & their circumcircles collinear upon inversion?

Consider an arbitrary logarithmic spiral of growth rate $q$ per angle $\theta$ and flair coefficient $b=\ln q/\theta$. Plot the spiral $z=e^{(b+i)\theta}$ and mark off the points that are equally ...
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Bettter way to map a vector fo the lower half of a symmetric matrix back into a vector

Assume a vector $v = [v_0, v_1, \dots, v_n]$ contains the values of a symmetric $n \times n$ matrix $M$. The map from $M$ to $v$ is $v_{k} = M_{ij}$ where $k = i(i+1) / 2 + j$ and $j \leq i$, in ...
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Avoiding inverse matrix calculation in Hessian approximation in Davidson-Flatcher-Powell method formula

My general concern is regarding denominator computation in this part $$\frac{s_ks_k^T}{y_k^Ts_k}\tag{1}$$ of the Hessian update in the method's algorithm/formula. Formula has taken from here : $$H_{...
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Inverse of nonsquare matrix product

I have a $n\times m$ matrix, $C$, and a $m\times m$ matrix, $B$. Is there any computational trick that allows me to calculate $(CBC^T)^{-1}$ without first calculating $Z=CBC^T$? Specifically, when $m&...
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How to inverse the laplace transform $\frac{1}{\cosh(5\sqrt{s})}$?

Let $X$ be a random variable with $ E[e^{-sX}]=$ $\frac{1}{\cosh(5\sqrt{s})} $ and density function $f$. How to give a formula for $f$?
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Does every mathematical operation has an inverse operation?

For example, we say that the addition and subtraction are inverse operations like that does each and every mathematical operation has an inverse operation?
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How to invert $f(x)=\dfrac{2^x}{1+2^x}$ [closed]

I know that if $$f(x)=\frac{2^x}{1+2^x}$$ then $$f^{-1}(x)=\log_2 \frac{x}{1-x} $$ How can I show this?
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1answer
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Approximating the inverse of a polynomial function for camera calibration

The method below to remove lens distortion from a camera was written more than ten years ago and i am trying to understand how the approximation works. ...
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1answer
41 views

Find the inverse of the function: $f(x)=x^9+x$

Let $f(x)=x^9+x$. Show that $f$ has an inverse and find the inverse. I don't seem to be able to find a way to start tackling this equation. Appreciate any tips on this question.
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Calculate inverse function

Is there a way to calculate the inverse of the following function? $$ s(t) =k\cdot \frac {e^{\alpha t} - 1}{e^{\beta t} - 1} $$
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How do I proof that $A=\sum\limits_{i=1}^{m}x_{i}x_{i}^{T} $ is invertible if and only if $X$ has full rank?

Show that $A=\sum\limits_{i=1}^mx_ix_i^T$ is invertible if and only if $x_1,\cdots,x_m$ span $\mathbb R^d$ for $x_i\in\mathbb R^d$. Here are my thoughts: If $A$ is invertible $Aw=0$ only has the ...
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What is the most efficient way to find the inverse of large matrix?

Let $A$ be a large square $(n+1) \times (n+1)$ invertible matrix, where $n \approx 1000$. $$A = \begin{bmatrix} -1 & 0 & 0 &\cdots & 0 & a_0\\ 1 & -1 & 0 &\cdots & ...
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1answer
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What must be true for $Ax = b$ to imply that $x = bA^{-1}$?

What must be true for $Ax = b$ to imply that $x = bA^{-1}$? Assume that $A$ is a matrix, and $x$ and $b$ are column vectors. 1) Is A singular or nonsingular? From the given (implied) equation, it ...
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1answer
48 views

What is the shortest way to find inverse of a matrix?

I know two methods to find the inverse of a matrix already:- Row and Column transformations $A^{-1}= \frac{Adj(A)}{Det(A)}$ I want to know if there's any shorter method to do so because these two ...
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1answer
79 views

Invertibility of the matrix whose elements are the cube of the distance of the indices.

I would like to prove, for any integer $n>1$, the invertibility of the $n\times n$ matrix $A$ whose elements are given by $A_{ij}=|i-j|^3$, where $i$ and $j$ are the indices. To be clearer, for ...
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Evaluate $|A – |A|\operatorname{adj}A|$ if $|A + |A|\operatorname{adj}A| = 0$ where A is a non-singular matrix of order 2

$A$ is a non-singular square matrix of order $2$ such that $$|A + |A|\operatorname{adj}A| = 0$$ where $\operatorname{adj}A$ represents adjoint of matrix $A$, and $|A|$ represents $\det(A)$ . ...
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Block identity matrix inversion

Let us consider the matrix $A \in \mathbb R^{2N \times 2N}$ defined as \begin{equation} A = \begin{pmatrix} I & I+\Lambda_{12} \\ I + \Lambda_{21} & I\end{pmatrix}, \end{equation} where $I$ ...
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1answer
35 views

Matrix relation for $E_{ij}(a)=I_n+ae_{ij}$

I want to show that for pairwise differen $i,j,k$ and the matrix $E_{ij}(a) = I_n + ae_{ij}\in \text{SL}_n(K)$ the following relation holds: $$E_{ij}(ab) = E_{ik}(a)E_{kj}(b)E_{ik}(a)^{-1}E_{kj}(b)^{-...
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1answer
57 views

Showing a bound on $x^\intercal A^{-1} x$ using $A$

Let $A \in \mathbb{R}^{n \times n}$ be symmetric and positive definite and fix vector $x_0 \in S^{n-1}$, which means $\|x_0\|_2 = 1$. We want to show $$x_0^\intercal A^{-1} x_0 \leq c$$ for a ...
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1answer
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Inverse of DiracDelta at 0 is 99/5?

When using Mathematica I've found an interesting result. InverseFunction[DiracDelta][0] == 99/5 (* returns True *) Or the inverse function of the DiracDelta ...
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A subring may have a different identity,why? [duplicate]

Let $R$ be a ring with unity $1_R$. Suppose $S$ is a subring but it does not contain $1_R$. But still it may contain a subring unity $1_S$. For example $R=M_2(\mathbb R)$ and $S$ be the set of all ...
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Modular inverse: why am I allowed to use the formula :$a^{-1}\mod p \equiv a^{p-2}\mod p$ [closed]

Why am I allowed to do this? Do you have a reference? $$a^{-1}\mod p \equiv a^{p-2}\mod p$$ Where do I get this? From here ...
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11 views

Proof for converting one inverse trigonometric function to another

I have tried to prove this result by checking the domain of two functions but every time I get entangled the last result so how to prove it
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207 views

Invertibility of elements in $A[x]$ with coefficients in the Jacobson radical

While solving an exercise about invertibility of elements in a polynomial ring, I came up with the following "proof" that a polynomial is invertible if its zeroth coefficient is invertible and all ...
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19 views

How to invert a multivariate series

Given 2 power series of 2 variables, $y_1(x_1,x_2)=\sum\limits_{m=0}^\infty\sum\limits_{l=0}^\infty a_{m,l}x_1^mx_2^l$ $y_2(x_1,x_2)=\sum\limits_{m=0}^\infty\sum\limits_{l=0}^\infty b_{m,l}x_1^mx_2^...
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3answers
78 views

Invertibility of infinite-dimensional matrix

I have a matrix $M \in \mathbb{R}^{n \times n}$ whose columns are linearly independent. Hence, $M$ is invertible. How to extend this conclusion to the case where $n$ is infinite? Specifically, ...

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