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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Inverse circulant matrix is circulant

I am trying to proof that the inverse of a circulant matrix is also circulant and had figured the best way to do it would be using the diagonalisation: $$ \begin{align*} C^{-1} &= (\frac{1}{n} F_{...
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Complex number Method verification for finding the sum of $\arctan 1+\arctan2 + \arctan3$

In the complex number method posted by a fellow MSE member here : https://math.stackexchange.com/a/272244/ , I think its not true always as what he/she did , assuming $\operatorname{Arg}$ represents ...
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Remarquable identities $f(n) = \frac{a^n}{(a-b)(a-c)} + \frac{b^n}{(b-a)(b-c)} + \frac{c^n}{(c-a)(c-b)}$

Let $n$ be an integer, and \begin{equation} f(n) = \frac{a^n}{(a-b)(a-c)} + \frac{b^n}{(b-a)(b-c)} + \frac{c^n}{(c-a)(c-b)} \end{equation} \begin{equation} g(n) = \frac{(bc)^n}{(a-b)(a-c)} + \frac{(ac)...
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Combing 2 Inverse Trig Equations into one expression

I am trying to solve for $x_0$ from the equation $tan^{-1}(x)-tan^{-1}(x_0)=t$. Do you have any suggestions?
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Why are algebraic equations sometimes displayed as inverses instead of solving for the variable in question?

As the title suggests, why are some algebraic equations displayed as inverses instead of equalling the direct variable we're trying to solve? For example, consider the d-spacing formulas used in ...
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Inverse of binary funcation

Is it possible to get the inverse function $f^{-1}(y)$ from the one below? Don't really know how to handle $\mathbf{1}_{\{ p(k)\leq x\}}$. $$ f(x)=\sum_{k=0}^\infty \frac{\lambda^k e^{-\lambda}}{k!}\...
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Method checking to prove the below statement regarding a function and its inverse

If a function "$f$" has a inverse and given that there is a point $x= a$ (lying in the domain of $f$) such that $f(a) = b$ , $a\neq b$ and $f^{-1}(a) = f(a)$ . Show that $f(x)$ is a self-...
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Range of $\cot^{-1}(x)$

S. L. Loney mentions the range of $\cot^{-1}(x)$ to be $[-\pi/2,0)\cup(0,\pi/2]$ whereas my textbook (my coaching institute package) mentions it to be $(0,\pi)$. May I know which one is the correct ...
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Solution to equation with moore-penrose inverse

I have a linear equation of the form $$ C = (I-AA^+)X $$ where my variable is $X$ and $A$ is an hermitian operator and $A^+$ is the pseudo inverse. Assuming that the determinant of $A$ is $0$ (or ...
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Domain and range of $f(x)=3\sin^{-1}(5x)$

I am trying to determine the domain of $$f(x)=3\sin^{-1}(5x).$$ Since the domain of $g(x)=\sin^{-1}(x)$ is given by the set $\{x\in\mathbb{R}:-1\leq x\leq 1\}$, I understand that the domain of $f$ ...
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Equality of product of matrices

I have been wondering if the following statement is corrrect? Assume that $A$ is $n \times m$ matrix where $m \leq n$ and rank of $A$ is $m$, $X$ and $B$ are $m \times m$ square matrix and it is ...
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Matrix identity involving inverse

Let matrix $Y \in \mathbb{S}^{n}$ be symmetric and positive definite (thus invertible) and $X \in \mathbb{R}^{n \times n}$ such that $Y - X X^T \succ 0$. I think that the following identity should ...
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Computing inverse elements of symbolic matrices with binary variables

I'm working with symmetric, symbolic matrices $A$ with real coefficients and linear binary variables like $$ A = \begin{pmatrix} 0.5x_0 & 0.3x_1+0.002x_2 & 0 & 0 \\ 0....
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Determine inverse matrix of $A = \begin{bmatrix} 1 & 2 & -1 \\ -3 & 1 & 2 \\ -2 & 2 & 1 \end{bmatrix}$

Could you give me your feedback ? Determine the inverse of the following Matrix: $$A = \begin{bmatrix} 1 & 2 & -1 \\ -3 & 1 & 2 \\ -2 & 2 & 1 \end{bmatrix}$$ We want to ...
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$A+USV^T$ is invertible iif $S^{-1}+V^TA^{-1}U$ is invertible

I was tasked to prove the Woodbury identity and as an intermediate step I need to show that $A+USV^T$ is invertible iif $S^{-1}+V^TA^{-1}U$ $U,V\in \mathbb{R}^{n\times m}$ and $A\in \mathbb{R}^{n\...
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Consider the matrix $H_n = (\frac{1}{i + j - 1})_{1 \leq i, j \leq n}$. Determine $H_2^{-1}$ and then show the given $H_3^{-1}$ is inverse of $H_3$

Could you give me your feedback ? Let $n \in \mathbb{N}$. We consider the matrix $$H_n = (\frac{1}{i + j - 1})_{1 \leq i, j \leq n}$$ It is interesting to observe that all entries of $H_n^{-1}$ are ...
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Show that if $B$ is the inverse matrix of $A^2$, then $AB$ is the inverse matrix of $A$

Let $A, B \in \mathbb{R}^{n \times n}$ be invertible matrices. Show: If $B$ is the inverse matrix of $A^2$, then $AB$ is the inverse matrix of $A$ $AA \cdot (B) = I$ then $A \cdot (AB) = I$ so ...
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finding the inverse of a function from $M_2(R).$

I am trying to prove that the given map below is a diffeomorphism and it is pretty clear to me that it is a bijection but I do not know how to show that the inverse of the given map is smooth? in fact ...
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Is it proper to speak of the inverse image of a set containing elements not in the image of the mapping?

This may be a duplicate of Inverse image of a subset of the codomain with elements without corresponding elements in domain , but the answer given seems to contradict what is intended by CH Edwards in ...
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1 answer
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Best way to compute $A^{-1}$ when the Cholesky decomposition $A=LL^T$ is known

Suppose $\mathbf{A}$ is symmetric positive definite, and that I have available the Cholesky decomposition of $\mathbf{A}=\mathbf{L}_A\mathbf{L}_A^T$. I want to know $\mathbf{A}^{-1}$. Which of the two ...
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1 answer
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Inverse of a function related area problem

My solution is based on the following diagram which I will illustrate in details Now ${A_1} = \frac{5}{4};{A_3} = 1$ $g\left( {2x} \right) = 2f\left( x \right)$ $g\left( 2 \right) = 2f\left( 1 \right)...
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2 answers
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Prove that any positive semi-definite matrix is "nearly" invertible.

A positive semi-definite matrix $\Sigma\in\mathbb{R}^{p\times p}$ is said to be $\eta$-invertible if there is an approximate inverse matrix $\Theta$ such that $$\max_{j,k}|(\Sigma\Theta-I)_{jk}|\leq\...
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Is this an incorrect application of Tutte's theorem of perfect matching for bipartite graphs?

This is an extract from a conference paper. It seems the authors are invoking Tutte's theorem (since [12] refers to the 1947 paper) to conclude that a matrix $J(x)$ with given numerical entries is ...
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If a random matrix converges to an invertible constant matrix $A$, does its inverse converge to $inv(A)$?

Let $B_n$ be a sequence of random matrices and $$ \underset{n\rightarrow \infty}{\text{plim}}(B_n) = A, $$ with A invertible. Does this imply that $$ \underset{n\rightarrow \infty}{\text{plim}}(B_n^{-...
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Inverse of a roto-translation matrix in 3D space

I want to create two roto-translation matrices. The first transforms point $P$ into point $P'$ by performing a translation $T=(x_t, y_t, z_t)$ and two rotations (one around the $x$ axis of $\alpha$ ...
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Inverse of $h(x) = 1 + x - \sqrt{1+2x}$

I want to compute the inverse of $$h(x) = 1 + x - \sqrt{1+2x}$$ for $x > 0$. To do so I have started from $$\begin{align}y = 1 + x -\sqrt{1+2x} \iff \\ y -1 = x -\sqrt{1+2x} \stackrel{y=1+2x>0}...
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Suppose that $dim(U) = dim (V)<\infty$ and let $T\in Hom(U,V)$. If $AT=\iota_V$ or $TA=\iota_U$, then $A$ is an isomorphism and $A^{-1}=T$.

I have a question regarding the proof of this corollary 2.11, which I have encountered in a course on Linear Algebra. As this is a corollary, I have a feeling that the proof should be somewhat obvious ...
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Inverse of Grassmann variables

Let $\theta$ be a Grassmann variable. I know that $1/\theta$ is not defined but $\frac{1}{1-\theta}=1+\theta$. My question is simple: is $\frac{1}{\bar{\theta}\theta}$ defined and if so how? My guess ...
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Division (by 0) and multiplication (as its inverse operation).

Division (by 0) and multiplication (as its inverse operation). $\frac{a}{b} = c \implies a = bc.$ $b (\frac{a}{b}) = a.$ If this is true, why is an explanation for the reason you cannot divide by zero ...
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Power series for $(A+\epsilon I)^{-1}$ for large $\epsilon$

I am trying to figure out a power series for $(A+\epsilon I)^{-1}$ when $A$ is an invertible matrix and $\epsilon$ is large. The Neumann series can be used when $\epsilon$ is small. Is there something ...
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Show that the multiplication operator $M_a$ is invertible in $L_p (\mathbb{R}^n) $ iff $a \neq 0$ almost everywhere and $(M_a)^{-1} = M_{a^{-1}}$

I mean, it's basically trivial, because $$M_a f = a(x) f(x)$$ And the only other operator $X$ to satisfy $X M_a f = M_a f X = If$ is $X = M_{a^{-1}}$, thus $(M_a)^{-1} = M_{a^{-1}}$, because $a (x) \...
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3 votes
3 answers
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Finding the Inverse of a Matrix using Row Operations

Problem: Let $A$ be the following matrix. Find $A^{-1}$. $$ \begin{bmatrix} -1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} $$ Answer: \begin{align*} \begin{bmatrix} -...
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Finding mean of normal distribution given probability between two endpoints

There is a well-known method for finding the mean of a normal distribution (given its variance) given the probability below a certain endpoint by normalizing the distribution: $$X \sim N(\mu, \sigma^2)...
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1 answer
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Inverting a function of 3 variables [duplicate]

We can define a stereographic projection by $$f(x_1,x_2,x_3)=\frac{x_1+ix_2}{1-x_3}$$ It is said the inverse of f can be computed to receive the general point where the line intersects the sphere. ...
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How to calculate inverse regression from a dataset

For a project, I need to program an algorithm to calculate the inverse function given by the form $$\ y=\frac{A}{x}+B $$ from a bivariate dataset. Does anyone know where I might find a description of ...
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What is coordinate matrix $[1_v]_{xy}$? And how can I show it is equivalent to composition of matrix and its inverse?

I am trying to solve the following problem from the previous year's exams of my Linear Algebra 1 (using FDLA by M. Gockenbach textbook): Suppose $V$ and $W$ are finite-dimensional nontrivial vector ...
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Formula for matrix inverse with non-commutative entries [duplicate]

I have a square matrix $A$ with elements $A_{i,j}\in\mathbb{A}$ where $\mathbb{A}$ is a ring with with addition ($+$) and multiplication operations ($\times$). The operation $\times$ is non-...
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Expansion of $\sinh^{-1}$ at $-\infty$

Excuse me if I'm being dense, but how do you derive $$\lim_{x \to -\infty} \sinh^{-1} (x) = -\infty$$ I have $$ \sinh^{-1}(x) = \log \left(x + \sqrt{x ^ 2 + 1}\right) = \log \left(x \left(1 + \...
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Invertibility of the Gram matrix of convex combination

I am struggling with this this question: Let assume two real valued matrices $A,B\in R^{w\times d}$, which $w>d$ and they both have full (column) rank. Now, I am interested to study invertibility ...
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Leading eigenvalue and Leontief inverse

Is it possible to get a Leontief inverse matrix $(I-C)^{-1}=I+C+C^2+....$ not equal to $I$ when the leading eigenvalue of matrix $C$, meaning the maximum of absolute eigenvalues, is zero? I am doing ...
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Are there applications for the inverse of the arc length of $ax^n$ and $a^x$? “Closed forms” found.

Based on: How to straighten a parabola? and Arc length of $x^n$ found using Hypergeometric function and series. Alternate representations and solution verification needed. Use: $$\text{ArcLength}(...
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Number of right inverses for surjective map between two sets.

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$, $B = \{x, y, z\}$, and define $f : A → B$ by $f(1) = f(3) = f(4) = y, f(2) = f(5)=x, f(6) = f(7) = z.$ How many functions $g : B → A$ satisfy $f ◦ g = id_B$? This is ...
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How radius of convergence of$~{3\over\sqrt{1-9x^2}}~$can be determined as$~{1\over 3}~$?

I want to prove the radius of convergence for the following is$~{1\over 3}~$ $${3\over\sqrt{1-9x^2}}\tag{1}$$ By the advice from@Kavi Rama Murthy, I thought the following. $$f(y):={1\over\sqrt{1+y}}=\...
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1 vote
1 answer
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Inverse of transformation matrix used to convert sphercial polar unit vectors to cartesian unit vectors

So I came across this question from physics stack exchange. I'd like you to see the first answer there. I am particularly interested in the conversion from spherical polar unit vectors to cartesian ...
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Question regarding identity matrix - $I_n$ and $I_m$ - Determinant ) - Prove B matrix is squared and invertible

question regarding Determinant of Identity matrix. I have a question regarding: A is $m_xn$, B is $n_xm$ $$AB=I_m$$ $$BA=I_n$$ I have to prove B is invertible and squared. I saw some solutions here ...
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2 answers
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Proof verification: If f is injective, f has a left inverse

I would appreciate verification of the following proof attempt, please. I suspect it is faulty at step 3, as I don't know whether I can use the existence of g as defined below as a premise. Statement:...
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1 vote
1 answer
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Inverse of a rational complex function and order of zeroes

If we have a complex function $f(z)$ that can be written as $$ f(z) = \frac{P(z)}{Q(z)} $$ and $P(z)$ has a single zero $z = z_0$ of order $n > 1$, I've read in ''Conformal field theory'' by P. Di ...
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Bounding the norm of an inverse matrix from above

In my situation I have an $s\times k$ integer matrix $X$ of rank $s$ and an $k\times k$ positive diagonal matrix $D$, where $s\leq k$. Say I can bound $||X||_{\infty}$, and I know $D$ precisely. Can I ...
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1 vote
1 answer
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Prove that $P$ is invertible.

Let $W$ be a subspace of $\Bbb R^m$ with dimension $n$. Let {$u_1$$,...,u_n$} and {$w_1$$,...,w_n$} be bases for $W$. Let $P=(p_{i,j})$ be an $n\times n$ matrix such that $$w_i=\sum_{j=1}^n p_{j,i}u_j=...
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  • 149
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Example of surjective function without right-inverse (without AoC)

As I understand from this question, without the axiom of choice, we can have surjective functions without right-inverse. Is that correct? Is there any such example of a surjective function where we ...
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