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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2
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17 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 ...
0
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1answer
19 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' \\ ...
0
votes
2answers
26 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
votes
2answers
50 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 $$f: A \mapsto A^{-1}$$ ...
2
votes
0answers
34 views

Inverse of two matrices multiplied

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 ...
0
votes
1answer
19 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 ...
0
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0answers
17 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} ...
0
votes
1answer
25 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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0answers
25 views

Laplace transformation with circuits

Im confused about the significance of inversing laplace transformations. What is the interpretation of s compared to t? Why is each Laplace transform only defined for some values of s? hopefully ...
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votes
0answers
25 views

Prove that a linear map has a left inverse if and only if the kernel of $L $ consists of the zero vector 0 [on hold]

Prove that a linear map has a left inverse if and only if the kernel of $L $ consists of the zero vector 0 alone. Prove that when $L$ has both a right left and a right inverse, then these inverses are ...
0
votes
1answer
399 views

relation between size of matrix and condition number

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
0
votes
0answers
15 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 ...
0
votes
1answer
10 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, ...
3
votes
2answers
183 views

How adjacency matrix shows that the graph have no cycles?

Let $G$ a directed graph and $A$ the corresponding adjacency matrix. Let denote the identity matrix with $I$. I've read in a wikipedia article, that the following statement is true. Question. Is it ...
0
votes
1answer
20 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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votes
2answers
447 views

Inverse functions and tangent line

Let $f(x) = \frac14x^3 + 12x + 6$ and let $y = f^{-1}(x)$ be the inverse function of $f$. Determine the $x$-coordinates of the two points on the graph of the inverse function where the tangent line is ...
0
votes
4answers
38 views

Can we show that $K=\tan(\frac{\theta_B}{2} + 45^\circ)$, given $\theta_B = \arctan(K) - \arctan(\frac{1}{K})$?

I am studying two separate technical documents which are about the same topic. I would like to know if they are defining certain two variables exactly the same. In the first document, it defines a ...
0
votes
4answers
51 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
24 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 ...
0
votes
2answers
59 views

Kalman filter innovation residual inversion

I'm trying to implement a Kalman filter in a computationally efficient way. The main issue is the inversion of the innovation residual: $$S=HPH^T+R$$ $$K=PH^TS^{-1}$$ My question is, can one assume ...
2
votes
1answer
935 views

Proof of Vandermonde Matrix Inverse Formula

I'm working through Exercise 40 from section 1.2.3 of Knuth's The Art of Computer Programming volume 1, but am finding myself unable to produce a rigorous proof, and the one here is suspect and not ...
2
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1answer
51 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
6answers
41 views

Finding the inverse of a number under a certain modulus

How does one get the inverse of 7 modulo 11? I know the answer is supposed to be 8, but have no idea how to reach or calculate that figure. Likewise, I have the same problem finding the inverse of 3 ...
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) ...
0
votes
1answer
88 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 ...
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votes
1answer
28 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
33 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) = ...
1
vote
1answer
21 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
51 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
16 views

Functions problem: surjectivity and direct and inverse image theory

I need some help with this problem, if sombody could give me any idea of how to solve it (not the solution itself, but it would be better) I will appreciate it: for a function $f: A → B$, prove $ ∀ Z ...
1
vote
1answer
15 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 ...
0
votes
0answers
10 views

inverse fourier transform of unit impulse function of omega

What is the inverse fourier transform of the unit impulse function of omega. Sorry I've not got the symbol in my phone. It Should looks like §(W).. Sorry for the special symbols.
0
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1answer
25 views

Inverse of elementary functions

which may be two right inverse of: 1) $h:\Re \rightarrow [0,\infty) $ defined by $h(x)=|x|$ 2) $k:\Re \rightarrow [1,\infty)$ defined by $k(x)= e^{x^2}$
1
vote
2answers
35 views

$\sqrt{4x -3}$ injective? Bijective? Inverse?

I'm shown part of the function $g(x) = \sqrt{4x-3}$. Is it injective? I said yes as per definition if $f(x) = f(y)$, then $x =y$. Is this right? Under what criteria is $g(x)$ bijective? For what ...
0
votes
1answer
61 views

Analytic inverse of $f(z) \neq 0, f(0) = 0, f'(z) \neq 0 $ within minimum modulus on boundary.

Suppose $f(z)$ is analytic on closed disk of radius $r$ and $f(0)=0$, $f'(z) \neq 0$. Show that $f$ has an analytic inverse on $\{|z| \leq m\}$ where $m$ is the minimum of $|f(z)|$ on $\{|z| = r\}$. ...
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votes
3answers
23 views

Verify matrix identity

The question: Show that if $A$, $B$, and $A+B$ are invertible matrices with the same size, then: $$A(A^{-1}+B^{-1})B(A+B)^{-1} = I$$ I began by multiplying the first $A$: ...
1
vote
1answer
57 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
3
votes
1answer
25 views

Inverse Laplace Transform with time delay and extra factor

I am attempting to solve a PDE $$y_{tt} = y_{xx}, -\infty < x < 0,\ t > 0$$ with boundary conditions $$ y_x(0,t) = k(t),\ y(x,t) \rightarrow 0\ \mbox{as}\ x \rightarrow -\infty,\ y(x,0) = 0,\ ...
0
votes
0answers
13 views

Closed form expression for the S-transform of a random variable?

I'm trying to compute the S-transform as described in this review article on random matrix theory (section 2.2.6). They define it as $\Sigma_X(\gamma) = -\frac{\gamma+1}{\gamma}\eta_X^{-1}(1+\gamma)$ ...
1
vote
2answers
38 views

Computing $\arctan x$ in terms of a certain collection of other functions

I know that $$\tan(x) = \frac{\sin(x)}{\cos(x)}.$$ Does this relationship hold in the inverse in any form? For example: atan(x) = asin(x) / acos(x), or atan(x) = acos(x) / asin(x), or atan(x) = ...
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votes
1answer
31 views

Derivation of matrix diagonalisation formula? [closed]

I can't seem to find a derivation of $A=P^TDP$ or an explanation of why this works or why it is important. I would be very grateful if someone could explain this, or perhaps give any useful lis about ...
0
votes
3answers
49 views

Inverse function of $x^x$

How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried ...
5
votes
1answer
139 views

Find Functions That Can Be Inverted from Their Sums

I have the following situation:$$ f_1(x_1) + f_1(x_2) + f_1(x_3) + \cdots + f_1(x_n) = c_1\\ f_2(x_1) + f_2(x_2) + f_2(x_3) + \cdots + f_2(x_n) = c_2\\ \vdots\\ f_n(x_1) + f_n(x_2) + f_n(x_3) + ...
1
vote
1answer
24 views

Showing topological properties of a function

Let $f$ be a function from a set $X$ into a set $Y$. prove: i) the function $f$ has an inverse if and only if $f$ is bijective ii) let $g_1$ and $g_2$ be functions from Y into X. If $g_1$ and $g_2$ ...
5
votes
5answers
133 views

Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$?

Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$ In other words, To show $f$ and $g$ are inverse, is it enough to show $(f\text{ o }g)(x)=x$? I have never witnessed a case in which the ...
0
votes
1answer
91 views

Find the inverse of a piece-wise continuous function

I have the following piecewise continuous function: $$f(x)=\begin{cases}3x+1,~x\gt 0\\2-x^2,~x\leq 0\end{cases}$$ and I need to find its right inverse. Thus far I got that ...
0
votes
0answers
7 views

inverse of semi infinite toeplitz matrix

I have a semi infinite toeplitz matrix of the form $ A=\left(\begin{array}{ccccc} A_{0} & A_{1} & 0 & 0 & \cdots\\ A_{-1} & A_{0} & A_{1} & 0 & \cdots\\ 0 & A_{-1} ...
3
votes
1answer
1k views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
0
votes
0answers
13 views

“Fix” an Inaccurate Matrix Inverse

I have $A$, a poorly-conditioned (but not singular) matrix. The matrix $B\approx A^{-1}$ (currently computed using double-precision Gaussian elimination), is rather further from the true $A^{-1}$ ...