Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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Norm of the inverse of a tridiagonal

Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric) $$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ ...
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13 views

Differentiating integral by substituting inverse function

I have the following cost function that I wish to minimize with respect to $\alpha$: ...
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1answer
36 views

Formal name for the coordinate values of the pushforward of the inverse metric on an embedded manifold?

What is the formal name of the following object: \begin{align}\tag{4} \Delta^{\alpha \beta} = \dfrac{\partial y^\alpha}{\partial x^m} g^{mn} \dfrac{\partial y^\beta}{\partial x^n} \end{align} where ...
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9 views

Inverse Variation Graph

How would I graph an inverse variation in which y varies inversely as x and y=2 when x=7? I know that I have to follow the xy formula. So far, I found xy=14, x(2)=14, x=7. How do I make a table and ...
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1answer
42 views

Inverting the Radial Distortion

Overview The problem is perhaps a very easy one for a trained mathematician. As I am not a mathematician, but instead a researcher in general problem solving, I am reaching out to those who know more ...
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57 views

Inverse of the Toeplitz matrix

I am working on the inverse of the sum of an identity matrix and a Toepltz matrix, and trying to find the formula for the (1,1) element of the inverse. For example, Assume $c$ is a nonzero constant, ...
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30 views

What is $dx/dF(x)$ where $F(.)$ is a continuous, increasing function.

I was wondering if it is possible to find $dx/dF(x)$, that is, the derivative of $x$ with respect to $F(x)$, which is an increasing, continuous function. Does it involve finding the derivative of the ...
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35 views

Inverse of $R^T D R$ where $R$: rectangular and $D$: diagonal

Is there any formula for the following triple product: $$(R^T D R)^{-1}$$ where $R$ is rectangular and $D$ is diagonal? The real situation is like this. I have the equation $$Ax = b$$ which is a ...
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2answers
39 views

How is $X(X^{\prime}X)^{-}X^{\prime}$ symmetric?

For a matrix $X$, a generalized inverse of $X$ is any matrix $Y$ such that $XYX = X$. We use $X^{-}$ to indicate a generalized inverse of $X$. Suppose $X$ is a matrix. $X^{\prime}$ denotes the ...
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3answers
24 views

Finding the inverse of an exponential function

I have this function: $$F_X(x) = \frac{3e^{2x}}{4} + \frac{3e^{4x}}{8} - 0.1$$ of which I am trying to find the inverse function, as in $u = F_X^{-1}(x)$. I made it to this form: $$u= ...
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1answer
34 views

Invertible matrix over a ring and its eigenvalues

Eigenvalues and invertible matrices for fields and vector spaces: Let $K$ be a field (so $K^n$ is a $K$-vector space) and let $A \in K^{n\times n}$ be an $n\times n$-matrix. Then we have the following ...
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5answers
47 views

Find the inverse function of $g(x)=(x-2)(x-4),\; x≥3$.

Find the inverse of the following function, stating its domain. $$ g(x) = (x-2)(x-4), \quad x≥3. $$ I try to find the inverse function, but I can't eliminate $x$ in my method.
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1answer
18 views

Left and Right Inverses with semigroups

Having the semigroup $(F,\circ)$ where $F=\{f: f: \mathbb{N}\to \mathbb{N}, \mathrm{Dom}(f) = \mathbb{N}\}$. The identity $e∈F$ is the function $e(n) = n$, define the function $g(n) = m$ if ...
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2answers
42 views

Inverse of a function $xe^x$

How should I proceed about finding the inverse of the function $xe^x$? I have been wondering about it for a long time and can't think of anything to do.
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53 views

Is there a closed form expression for $(A^T\Sigma A)^{-1}$ when $A$ is not square?

I need to find the inverse of the matrix $A^T\Sigma A$. Matrix $A$ has dimensions $5\times 2$. Matrix $\Sigma$ has dimensions $5\times 5$, and it is symmetric and positive-definite. I need to ...
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1answer
59 views

$f '(x) = -f(x)$ and $f(1) = 1$, Solve for $f(2)$

I am honestly not even sure how to start this problem... My first thought was that $f(2) = 2$ ... But now I don't even know where to go from there.
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23 views

How do you find the inverse of a multivariable function?

In 1D variable calculus, you have a nice theorem that says: Suppose $f$ is differentiable and has an inverse on $I$. Suppose $x_o \in I$ and $f'(x_0) \neq 0$. Let $y_o = f(x_o)$, then ...
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2answers
48 views

Can I use eigenvalues to find the inverse of a vector?

I have two 1D matrices (say dimension 1xn) called A and B. Multiplying these: A . B = M. Where M is a scalar. Knowing B and M, can I find A? One cannot take the inverse of a vector, but is it ...
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1answer
26 views

Inversion of $z$-transform using partial fraction decomposition

I want to inverse a $z$-transform of this general form $$X(z) = \frac{b_0 + b_1z^{-1}+\cdots+b_Mz^{-M}}{a_0 + a_1z^{-1}+\cdots+a_Nz^{-N}}$$ where $M$ < $N$. In order to do this, I use partial ...
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1answer
27 views

What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
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0answers
30 views

Inverse functions multivalued or not?

The square root of $y$ is usually defined as the positive solution $x$ to $y=x^2$, so the negative variant is not considered. In the same way, the inverse cosinus and sinus give the solution on ...
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1answer
40 views

Inverse of a matrix and its transpose

I'm trying to figure out why the calculation below works. I do know that $(A^T)^{-1} = (A^{-1})^T$. The matrix A = $\begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1 ...
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77 views

Multiplicative Inverse Element in $\mathbb{Q}[\sqrt[3]{2}]$

So elements of this ring look like $$a+b\sqrt[3]{2}+c\sqrt[3]{4}$$ If I want to find the multiplicative inverse element for the above general element, then I'm trying to find $x,y,z\in\mathbb{Q}$ such ...
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42 views

If $f(z)= \frac 1z $ be defined and analytic on region $ |z| \gt 1 $ in $ \Bbb C $ then can we find an entire function $g$ such that :

$g$ should be such that $f(z)=g(z)$ on $ |z| \gt 1$ in $\Bbb C $. Now,Can we plainly apply uniqueness theorem and say that such a function $g$ can not exist?
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1answer
30 views

Inverse of a set of ordered pairs.

An exam ask me the following question. Let $r=\{(x,y) \ | \ x \in [-1,1] \ \text{and} \ y=x^2\}$, is the following statement true? $$r^{-1}=\{(x,y) \ | \ x \in [0,1] \ \text{and} \ y=\pm\sqrt{|x|} ...
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5answers
48 views

Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$

Let E and F be two sets and $f: E \to F $ be a function, and $X, Y \subset F$. Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ My answer: Let $y \in X$, then $f^{-1}(y) \in ...
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0answers
22 views

Inverse of a constant function on an open set

I was working on holomorphic functions and Riemann surfaces, and I was wondering about the inverse of a constant function: Let $f:U\rightarrow V$ be a holomorphic function between two Riemann ...
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2answers
37 views

Is this definition correct for the inverse of a function?

Is this definition correct for the inverse of a function? Let $f:X\to Y$ be a function. The inverse of $f$ is the function $g:Y\to X$ such that $g\circ f=i_X$ and $f\circ g=i_Y$. We denote the ...
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1answer
22 views

convert the inverse of sum of two hermitian matrices into sum of two or more matrices.

I want to convert the inverse of sum of two hermitian matrices into sum of two or more matrices. I mean I want to simplify the bellow equation in a way that not to have inverse of sum of matrices any ...
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2answers
34 views

Finding inverses of a function which maps ordered pairs of positive integers onto the positive integers.

The function $f(x,y) = \frac{(x+y-1)(x+y-2)}{2} + y $ is a bijection which maps ordered pairs of positive integers onto the positive integers. I would like to find the functions $g$ and $h$ such that ...
2
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2answers
52 views

Homeomorphism between the set of invertible matrices and itself

Consider the set of invertible $n \times n$-matrices $GL_n(\mathbb{R}) = \{A \in M_{n \times n}(R) \mid A\text{ is invertible}\}$. I now want to prove that the transformation $$f: A \mapsto A^{-1}$$ ...
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0answers
161 views

Inverse of two matrices multiplied [closed]

I've been asked to find the inverse of $AB$ where $A$ and $B$ are: $$A=\begin{bmatrix}5 & 3 \\4 & 2\end{bmatrix}$$ $$B=\begin{bmatrix}2 & -3 \\1 & 3\end{bmatrix}$$ My answer: What I ...
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0answers
20 views

Solve $x$ in the equation: $a\cdot \textrm{arctanh} [b + a \cdot x] - c \cdot \textrm{arctanh} [d + c \cdot x] = e$

How to solve $x$ in the equation: $a\cdot \textrm{arctanh} [b + a \cdot x] - c \cdot \textrm{arctanh} [d + c \cdot x] = e$, where $\textrm{arctanh}(x) = \frac{1}{2} \log \left(\frac{1+x}{1-x} ...
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0answers
17 views

Question about inverse of CDF being a real analytic function

Let F: [0,a] -> [0,1] be a continuous, strictly increasing CDF. Assume also F admists a continuous, positive pdf f. Now define the inverse function h(x) as F(h(x))=x. Is h real analytic? If not, what ...
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1answer
27 views

Representation of the inverse of an variance-covariance matrix $\hat{\Sigma}^{-1}$

Given $T$ observed vectors $x_i\in\mathbb{R}^N, i\in\{1,\ldots,T\}$. Define $\hat{\Sigma}$ as the corresponding empirical covariance-matrix of the Observations $X=\left(\begin{array}{c} x_1' \\ ...
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1answer
36 views

Computing the inverse of linear transformations using matrices

For each of the following linear transformations T, determine whether T is invertible, and compute T-1 if it exists. (a) T: P2(R) $\to$ P2(R) defined by T(f(x)) = f ''(x) + 2 f '(x) - f(x). My ...
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1answer
11 views

Calculating the inverse of a continuous map for a certain interval in order to calculate the Perron-Frobenius operator.

Suppose we are observing chaotic continuos maps, the Perron-Frobenius operator $P$ satisfies: $P\phi_{n}(t) = \frac{d}{dt} \int_{f^{-1}([a,t])} \phi(x)dx$ I don't understand how for the shift map, ...
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1answer
22 views

What is the adjoint of an inverse matrix? [duplicate]

What is the adjoint of an inverse matrix? Is $(T^{-1})^{*} = (T^{*})^{-1}$?
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1answer
27 views

Permutations, compositions and associativity properties

Let n be a postive integer, and let σ : {1, . . . , n} → {1, . . . , n} be a one-to-one and onto map. Then σ is called a permutation on n elements. The set of all permutations on n elements is denoted ...
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4answers
54 views

Let $T$ be a linear transformation on a vector space $V$ ($\textrm{dim}\ V = n$). If $\textrm{rank}\ (T^2) = n$, is $T$ invertible?

For a linear transformation $T$ on a finite dimensional vector space $V$ ($\textrm{dim}\ V = n$). If $\textrm{rank}\ (T^2) = n$, is $T$ invertible? Also, is it guaranteed to have an eigenvalue?
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1answer
25 views

Series expansions of inverse polynomials

Suppose one is given a strictly monotonous polynomial, $$f(x) = \sum_{n=0}^N a_n x^n$$ So that for a given $y$ there exists a single real $x=f^{-1}(y)$. It would be nice* to be able to calculate the ...
2
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1answer
53 views

Inverse of $f(x) = 3x + \cos(x)$

Was hoping someone could help me find the inverse of $f(x) = 3x + \cos(x)$ The steps I took were: $y = 3x + \cos(x)$ $x = 3y + \cos(y)$ $x - 3y = \cos(y)$ $\arccos(x-3y) = y $ But I still have a ...
0
votes
1answer
23 views

Find $f^{-1}(g(x))$ if $f(x) = 2x + 1$ and $g(x) = x^{2}$

Question: Let $f$ and $g$ be defined as: $$f(x) = 2x + 1, ~~~~x \in \mathbb{R}$$ $$g(x) = x^{2}, ~~~~~~~~~~~~x \in \mathbb{R}$$ Find a) $~~f^{-1}(x)$ b) $~~f(g(x))$ c) $~~g(f(x))$ d) ...
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1answer
30 views

Why do the columns of the inverse of a matrix (defined as a linear operator) form an orthogonal basis in an inner product space?

Let V be a vector space over C and W be an inner product space over C with inner product <., .> and T:V --> W be a linear transformation. Find an orthogonal basis for V = R^3 with the inner product ...
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2answers
35 views

How to show $\text{rref }[\left.A\right|AB]=[\left.I_n\right|B]$?

For invertible $A^{n\times n}, B^{n\times n}$, how do I show that $\text{rref }[\left.A\right|AB]=[\left.I_n\right|B]?$ Tentatively: $\text{rref ...
2
votes
2answers
30 views

$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\frac{d}{dy}[\Phi(\Phi^{-1}(y))]}$?

If $\Phi(y)$ is a monotonic decreasing function is true that $$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\Phi'(\Phi^{-1}(y))}$$ If so, how? It works for $y = \Phi(x) = e^{-x}, \quad \Phi^{-1}(y) = ...
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1answer
23 views

how to do inverse laplace of $(s^2+1)/s^4$?

how to do the inverse laplace of $(s^2+1)/s^4$? the answer is $(t^3/6)+t$ but I do not know how to derive it.
2
votes
3answers
57 views

Find the inverse of $f(x,y) = (x+3y,3x+y)$

Given the function $f : \mathbb{R}^2 \to \mathbb{R}^2$ as $f(x,y) = (x+3y,3x+y)$. Find $f^{-1}$ .( Assume $f$ is a bijection) I know how to find $f^{-1} (x) = (3x+2)$ or anything with one ...
0
votes
1answer
94 views

Inverse of $f(x) = a \left(1 + \frac{c}{(1+x^b)^{-\frac{1}{b}} - c}\right) \cdot (1+x^{-b})^\frac{1}{b}$?

How can one find the inverse of $$ f(x) = \mathrm{a} \left(1 + \frac{\mathrm{c}}{(1+x^\mathrm{b})^{-\frac{1}{\mathrm{b}}} - \mathrm{c}}\right) \cdot (1+x^{-\mathrm{b}})^\frac{1}{\mathrm{b}} $$ with ...
1
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1answer
24 views

Inverse function table

I am required to create a table of values (like the one above) for h-1(x). Because x is ordered, i am just wondering, would the two tables would be identical? I just feel a little insulted that's ...