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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In what condition we have $(K^{-1})^\ast = (K^\ast)^{-1}$?

Suppose $X$ $Y$ are two finite dimensional Hilbert space. Assume $K$: $X\to Y$ is linear. My question is, in what condition of $K$ that $$(K^{-1})^\ast = (K^\ast)^{-1}?$$
4
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2answers
71 views

Is this matrix invertible?

I have been working on a proof and am stuck with showing that the below matrix is invertible. I am not interested in the explicit inverse, only showing it has a nonzero determinant as the existence of ...
2
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1answer
38 views

How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.

A while ago, I found that: $$f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt}.$$ I managed to prove it for a few functions, and I believe that it may be the case for all ...
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16 views

Multiplicative inverse of a (non-prime) field

We study field extensions. Two classes widely introduced to newbies of Galois Theory are (isomorphic to, though they may not appear in this form) $\mathbb{Q}^n$ and $\mathbb{F}_p^n$ (for some prime ...
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42 views

Inverse of partitioned matrices [on hold]

A matrix of the form $$A=\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix}$$ is said to be block upper triangular. Assume that $A_{11}$ is $p \times p$, $A_{22}$ is $q \times q$ and ...
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3answers
117 views

Harder-Than-Seems Inverse of $f(x)=x^3-x-12$?

This may seem simple but I have had long days of frustration with finding the inverse of this: $$f(x)=x^3-x-12.$$ I got this on some homework and it did not ask for the inverse. However I wanted to ...
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3answers
792 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
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2answers
55 views

Clarification of definition of “inverse” with quaternions

From what I understand, the inverse of a matrix only exists if the matrix is square. I recently learned however that the inverse of a quaternion is the quaternion vector (1xn dimensions) where each ...
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2answers
29 views

Does for $T \in B(X)$ with $\|T\|>1$ exist $T^{-1}$?

Is it true if $\|T\|>1$, where $T \in B(X)$ for some Banach space $X$, then $T^{-1}$ exists? I suppose that for $\|T\|=1$ this isn't true? Because, if we suppose that inverse exists for such ...
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2answers
37 views

inverse a function with exponential and first degree polynom

I need some help to inverse this function: $$ y = a(e^{bx}-1) + cx + d $$ with $y(0)=d$ and $y(k)=0$ where $k$ is a constant. I don't know how to proceed. Thanks.
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56 views

How to find the inverse of $f(x)=2x - x^2$? [closed]

What is the inverse of $f(x)=2x - x^2$ in the domain (0,1) and the range (0,1)?
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1answer
54 views

Find the inverse function of $ f (x ) = x^2 - x - 2$

Find the inverse function of $ f (x ) = x^2 - x - 2$, where x is equal to or larger than 1/2. I tried to express it in form of $ (x - 1 )^2 = y + 2 $, but this is not true as the term in the middle ...
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0answers
21 views

Why does the inversion of this circle give a horizontal line y=i/2?

Inverting the circle centered at $(0,-i)$ with radius 1, gives the horizontal line $y = \frac{i}{2}$, but why does it have to be horizontal - Why not another straight line passing through the ...
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1answer
37 views

Inverse and composite functions [closed]

If $f(x)=\frac{x}{1-√x}$, $x≥0$ and $g(x)=3x+1, $ Solve the equation $(f^{-1}\circ g)(x)=9/16$. Hint:do not attempt to find $f^{-1}(x)$.
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3answers
45 views

Obtaining a Non-Singular Matrix from a Singular one by Perturbation

In a paper "http://www.math.cornell.edu/~nussbaum/papers/08-1.pdf" (page 264 Lemma 2) I encountered the following way of obtaining an invertible (non-singular) matrix from a non-invertible (singular) ...
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1answer
43 views

Understanding $F \circ \phi^{-1}$ in differential geometry

I am struggling with a question in elementary differential geometry. I thought I understood the basics until I read page 20 of The Geometry of Physics by T. Frankel. Suppose we have a manifold of ...
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2answers
30 views

Prove $xf(x) - \int_{0}^{x} f(t) \,dt = \int_{f(0)}^{f(x)} g(t) \,dt, g(x) = f^{-1}(x), \forall f(x)$?

I have proved it for all functions of the form $f(x) = x^{a}, \forall a, x,$ but I am not sure if it is true for all functions. Proposition $$xf(x) - \int_{0}^{x} f(t) \,dt = \int_{f(0)}^{f(x)} ...
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0answers
20 views

Closest line to point after non-linear map

I have a map on a vector space $M(\vec{r})$, defined as below. All components (vectors, matrices, everything) are reals in the unit range $[0,1]$. The map $M(\vec r)$ is defined as the sum of an ...
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0answers
17 views

How to use the condition number to determine whether a matrix is easily inverted?

I have an enormous covariance matrix, but I don't know if it is feasible to take its inverse. So, I thought about finding its condition number, which would help give insight into how easy it might ...
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1answer
40 views

Inverse of a special matrix 3

Let $\mathbf{V}$ be a $2n-1$ by $2n-1$ [symmetric positive definite] matrix with a known inverse and define $\mathbf{A}=[[\mathbf{D},\mathbf{0}]',\mathbf{I}]$ where $\mathbf{D}$ is a diagonal matrix ...
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1answer
21 views

Inverse of sum of two functions in terms of individual inverse functions

When we can express the inverse of sum of two functions for example $f=f_1+f_2$ in terms of inverse of two functions $(f_1^{-1},f_2^{-1})$?
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50 views

The inverse of x!

what is the inverse of a factorial function? Its is not continuous but is modeled by the gamma function which is continuous so must have a inverse any research leads to the inverse gamma function that ...
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1answer
40 views

$h_1,\ldots,h_n$ are linearly independent if and only if $A$ is invertible

Show that $h_1,\ldots,h_n$ are linearly independent in $\mathscr{H}$ (Hilbert space) if and only if the matrix $A$, defined by $A=[a_{ij}]$ where $a_{ij}=\langle h_j, h_i \rangle$, is invertible. ...
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5answers
81 views

Is it necessary for a statement to have an inverse in propositional logic?

I know that it may be rather self-evident that every statement must possess an inverse, however, is there a way to prove this in propositional calculus or is it considered an axiom? (Note: By the ...
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1answer
24 views

Multiplicative inverse of polynomial modulus an integer

How do you calculate the multiplicative inverse of a polynomial mod a monomial/integer?The specific questions are: Find the multiplicative inverse of 1) x+1 mod 3 2) x^2+x-1 mod 3 3) x^2+x-1 mod 32 I ...
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20 views

numerical stability - matrix inversion

i am self-studying regression in Kevin Murphy's book (A probabilistic perspectivce), and i came across the following statement, which seems important, but was not further explained. What is the reason ...
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1answer
35 views

How many different functions are there that are equal to their own inverse? [duplicate]

I know that functions can be their own inverse such as $f(x)=x$ however I thought there were only two $f(x)=x$ and $f(x)=-x$. Is there more?
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1answer
62 views

How to solve $x=\sin^{-1}(\frac{1}{2\sqrt{x}})$

I was setting a question when I came across a problem. The question was: Suppose I have a function $y=e^{1+\cos(x)+\sqrt{x}}$. (A) Locate its turning points by taking derivatives and sketch its graph ...
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41 views

Verify if 2 functions are inverse to each other

According this site, 2 functions $f$ and $g$ are the inverse function of each other, only if both $(f \circ g) (x) = x$ and $(g \circ f) (x) = x$ are true. Is it really necessary to prove both of ...
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5answers
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Functions that are their own inversion.

What are the functions that are their own inverse? (thus functions where $ f(f(x)) = x $ for a large domain) I always thought there were only 4: $f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ ...
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0answers
40 views

Inverse of sum of 3 matrices

I need a way to compute the inverse of the sum of three matrices: $(A + BB^T + \beta I)^{-1} $ where $I$ is identity and $\beta$ is a constant. I am not very familiar with linear algebra, but a ...
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2answers
39 views

Using Least Squares to calculate a matrix in an equation.

I have two sets of vectors $v_i$ and $w_i$, in some $d$ dimensional space. I know that $v_i \approx M \cdot w_i$ for all i. I.e., I know that the $v$ vectors are a linear transformation of the $w$ ...
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2answers
34 views

Existence of Linear Transformation between 3D line and 2D line

I am wondering if there exists an invertible linear transformation between a line segment in 3D space and a line segment in 2D space. Basically, the red line above could be represented by the ...
0
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1answer
31 views

How do I solve for A in the matrix equation $A - B(A./C) = D$?

I've got $A - B(A./C) = D$, and I want to solve for $A$.* $A$ is an unknown 2x1 vector, $B$ is a known 2x2 matrix, $C$ is a known 2x1 vector, and $D$ is a known 2x1 vector. *The notation $A./C$ ...
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Matrix inverses - Why are they derived the way they are?

Note that this is not a question of how, but why. I know the mechanics of it, but this is the first thing i've come across that truly seems like magic, rather than a rigorous mathematical process. ...
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2answers
50 views

Finding the inverse of the function $f(k, x) = k^{x}x.$

Recently, I have been looking at the function $f(x) = e^{x}x,$ where its inverse is the Lambert W function. I was intrigued by the fact that it is rather hard to calculate its solution, in comparison ...
2
votes
3answers
148 views

Find a constant so matrix is invertible

I am doing some exercises from my Linear Algebra textbook and i have come across an exercise which I don't quite understand. Every exercise is graded with numbers from [1] to [5]. [1] is meant to be ...
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33 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz [duplicate]

I have asked this question on mathoverflow also. (my question, I wasn't sure if its ok ask at another similar forum, on stack exchange, but I hope it would reach more people). It is well known how to ...
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2answers
124 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz

It is well known how to solve a Toeplitz system Ax = b, of a matrix A, n x n elements, ...
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1answer
33 views

Is it okay to perform the same row operation twice on opposite rows?

I am trying to find the inverse of the following matrix: 1 2 3 2 1 4 1 0 2 I draw the identity matrix next to it and start performing row operations. ...
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1answer
25 views

Inverse Permutations from $S_7$

Would someone mind giving an explanation of how to find the inverse permutation of: $(1 2 3 5 7)^{-1}$ in $S_7$? I am not quite understanding how to do this.
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1answer
70 views

Finding a binary operation on $\{1, \dots, n\}$ so that each $k$ has exactly $k - 1$ left inverses

What is an example of a binary operation on the set $\{1, \dots, n\}$ so that each element $k \in \{1, \dots, n\}$ has respectively $k-1$ left inverses? I have been trying various combinations ...
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4answers
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If $A$ is a $3\times3$ Matrics Then $\left |(2A)^{-1} \right |=?$ [closed]

If $A$ is a $3\times3$ matrics.And $\left | A \right | = -7$.Then what's the value of $\left |(2A)^{-1} \right |$ Please help to do this math easily.I tried a lot but still no idea come into my ...
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3answers
51 views

Slopes of inverse functions

I have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ...
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2answers
29 views

Choosing the right sign for inverse functions?

If I have to find an inverse function and through the algebra I get a $\pm$ sign how do I know which one to choose from if its in a given interval? For example a question asks: The function ...
0
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1answer
24 views

Is there an explicit formula for $\left(xx^T\right)^{-1}$ with $x\in\mathbb{R}^n\setminus\left\{0\right\}$?

Let $x\in\mathbb{R}^n\setminus\left\{0\right\}$. Obviously, $$A:=xx^T$$ is symmetric and positive definite. Hence, $A$ is invertible. Can we find an explicit formula for $A^{-1}$?
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4answers
133 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
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0answers
49 views

Inverse Laplace Transform using Hetnarski's Algorithm

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
3
votes
1answer
53 views

Inverse Laplace transform of $\operatorname{arccot}(s)$, $\arctan(s)$

How would one find inverse Laplace transforms of $\operatorname{arccot}(s)$ or of $\arctan(s)$ without knowing in advance that this is related to $\dfrac{\sin x}{x}$?
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1answer
33 views

Find inverse $f^{-1}$ of a function $f(x,y)=(x-y,x-10y)$ [duplicate]

I know how to find inverse function if the given function is in the explicit form. Could someone show on this example how to find $f^{-1}$? Thanks for replies.