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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Finding the inverse of a function involving |x|

I need to find the inverse of the following function $ f:(-1,1) \rightarrow \mathbb{R} $ $ f(x) = \dfrac{x}{1-|{x}|} $ How do I deal with the absolute value here? Thanks
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2answers
515 views

Calculating the trace of the product of two matrices

I have to calculated $\mbox{trace}(A^{-1}B)$ where $A$ is a symmetric positive definite matrix and $B$ is a symmetric matrix, very sparse with only two elements non zero. I want to find a way that I ...
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202 views

Inverse of identity plus scalar multiple of matrix

Given the matrix $M = ( I + \alpha D P )$, where $I$ is the nxn identity, $D$ is nxn symmetric and invertible, $P$ is nxn symmetric but not always invertible, and $\alpha$ is a scalar, is there a ...
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37 views

General formula for the inverse of the symmetric matrix $X$ defined as $a^{x_{ij}}$

Let $X$ be a $N\times N$ symmetric matrix with strictly positive entries $x_{ij}$. The inverse of $X$ is known. Let $0 < a < 1$ be a real number. Finally define $M$ as the matrix with entries ...
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1answer
105 views

Efficient diagonal update of matrix inverse

I am computing $(kI + A)^{-1}$ in an iterative algorithm where $k$ changes in each iteration. $I$ is an $n$-by-$n$ identity matrix, $A$ is an $n$-by-$n$ precomputed symmetric positive-definite matrix. ...
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2answers
31 views

Is inverse image of a point a set?

This is purely a matter of notation. Suppose $f(x) = x^2$. We say that $f(2) = 4$. We also say that $f(\{2\}) = \{4\}$. We can also say that $f^{-1}(0) = 0$ without much grief because there exists a ...
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3answers
101 views

What is the inverse of $f(x)=\frac{e^x+e^{-x}}{e^x-e^{-x}}$?

please help me to find out the inverse this function, $$f(x)=\frac{e^x+e^{-x}}{e^x-e^{-x}}$$ I know that, let $$y=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$$ and if I find $x=\cdots$ then that is the ...
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1answer
75 views

Inverse Laplace transform of $\large \frac{1}{s^2-As^{1.5}}$

Title says it all. How do I go about finding inverse Laplace transform of that expression? If it were complete exponents, I would have used partial fractions. But what to do with non integer ...
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1answer
40 views

Why is there a left inverse for an injective Function with the empty set as domain?

The fact that a function is injective is equivalent to the fact that there is a left inverse. Now consider $f:\mathbb{∅}\to \mathbb{A}$ where $\mathbb{A}$ is non-empty. Wouldn't the left inverse be ...
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41 views

Inverse laplace transform of complicated function

I have a function: $f(s)=\dfrac{(-HT/s)e^{-x*\sqrt{a/s}}}{\sqrt{a/s}+He^{-x*\sqrt{a/s}}}$ where s is frequency domain variable and H,T,a,x can be regarded as constants. How do I find inverse Laplace ...
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Inverse of function, containing a fraction

This is basic, I know, but I cannot seem to come up with the right answer. Find the inverse of the function: $$f(x)= \frac3{x+1}$$ My steps: 1. Convert f(x) to y $$y = \frac3{x+1}$$ Switch places ...
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45 views

How to solve the following equality

Is it possible to solve the following equation analytically for $\beta$: $$y'(A+\beta B)^{-1}y = \alpha,$$ where $A$ and $B$ are both positive-semidefinite and symmetric matrices (essentially, some ...
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2answers
401 views

Can someone please help with my inverse function and sets discrete math problem?

To save me some time writing everything out in latex, I'm adding a picture of the question and Ill try to explain what I understand for the problem. Just a heads up, I'm really not sure how to do this ...
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1answer
49 views

Exercise about linear operator

For $X$ Banach, I have to show that if $T\in\mathfrak{L}(X)$ and $||T||_{\mathfrak{L}(X)}<1$ then exists $(I-T)^{-1}$ and $$ (I-T)^{-1}=\sum_{n=0}^\infty T^n. $$ For the existence of $(I-T)^{-1}$ ...
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218 views

Is there a meaningful pseudo-inverse of a singular projection matrix?

Hello linear algebra experts. In my research I'd like to solve (or approximate) for B, in the form $ A = GBG $ where A and B are symmetric, square matrices and G is a symmetric, square, singular ...
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29 views

Chain rule with inverse function

In a proof, my professor shows: $ s = g^{-1}(u) $ $ ds = \frac{dg^{-1}(u)}{du} du $ , by the chain rule If I were to apply the chain rule to calculate ds, I would not get the du in the denominator. ...
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95 views

Find the inverse of a matrix in $GL(2\,,\, \Bbb Z_{11})$.

What are the necessary steps and reasoning for calculating the following matrix in GL(2,$\Bbb Z_{11}$): $M = \begin{pmatrix} 2&6 \\3&5 \end{pmatrix}$. I found the answer to be ...
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4answers
97 views

Let $f: A\rightarrow B$ and $g: B\rightarrow C$ be invertible maps, show that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

I am working on the following problem for my abstract algebra class, and I wanted to get some feed back to see if I am on the right track. Here is what I have so far. Let $f: A\rightarrow B$ and $g: ...
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1answer
19 views

Finding Inverse of Function With Two Instances of X

I need to find $f^{-1}(2)$ where $f(x) = 2 + x^2 + tan(πx/2)$ I know can substitute $f(x)$ with $y$ and swap $x$ and $y$: $$x = 2 + y^2 + tan(πy/2)$$ But I'm having trouble eliminating the tangent: ...
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4answers
125 views

Inverse of $(e^x - e^{-x})/2$

What is the inverse of the function $f(x)=\frac{e^x - e^{-x}}2$? I tried replacing $e^x$ by a variable but I still can't get it.
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2answers
58 views

is (I+P) invertible when row sum in P = 0

I have a $n$x$n$ matrix P where the sum of each row = 0 (the individual entries are real but can be negative). Clearly P is not invertible. Can we show that I+P is invertible? thanks
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30 views

Finding the integral of an inverse cosine function?

I've just been having trouble with this question: "Differentiate $xcos^{-1}x$ and hence find the integral of $cos^{-1}x$. Hint: Try using the substitution $u=1-x^2$." Finding the derivative wasn't ...
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52 views

Inverse of the sum of two orthogonal projections

I am trying to find out, if there is a formula for finding the inverse of the sum of two orthogonal projections. So basically my questions is: If $\left[\mathbf{A},\mathbf{B}\right]$ is full rank, ...
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78 views

Suppose R is an integral domain. Prove that $(a)=(b)$ if and only if $b = ua$ where $u$ is in $R^\times$

I am lost on this one. I'm still new to ring theory, as we're only a couple weeks into the course, but it's already well over my head. I know that $R$ is an integral domain, so the additive and ...
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1answer
135 views

Inverse of $f(x)=\sin(x)+x$

What is the inverse of $$f(x)=\sin(x)+x.$$ I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet. What about $$f(x)=\sin(a \cdot x)+x$$ where ...
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1answer
47 views

Expressing an inverse trig function?

I just need a little help with this question: "Express cos$y$ in terms of cos $y/2$ and hence show that tan$^{-1} sqrt[(1-x)/(1+x)] = 1/2$ cos$^{-1}x$, for $0<x<1$." I can do the first part, ...
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1answer
26 views

Derivation of inner variations

In Giaquinta's and Hildebrandt's 1996, "Calculus of Variations 1", pages 147-148, they develop the definition of inner variations. They first fix $\lambda\in ...
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269 views

inverse of quadratic matrix form

I have an expression of the form: $ACA′$ where C is an invertible, symmetric and positive definite matrix. I'm trying to figure out if the expression above is invertible (or what additional ...
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1answer
480 views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
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75 views

Is there a name for an algebraic structure like this?

I'm self studying abstract algebra. I see that in rings there's no requirement for a multiplicative inverse. Is there something similar except with no requirement for an additive inverse. For ...
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1answer
104 views

Increase the diagonal entries of a positive definite matrix

Assume that we have a positive definite matrix $C$, and a positive definite diagonal matrix $\Lambda$. Are all the diagonal entries of $(C + \Lambda)^{-1}$ smaller than those of $C^{-1}$? In other ...
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39 views

Matrix problem involving an equation.

Please could you help me with the below question. There are three parts, and all of my working is displayed! Thank you in advance, kind stranger. For an integer n, real numbers a,b,c and an nxn ...
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0answers
40 views

Error bounds in representing a vector using a truncated Moore-Penrose biorthogonal basis

I was reading and trying to reproduce the results in the arXiv preprint of Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression by Asaf ...
2
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8answers
91 views

Find the inverse of the following matrix.

How can I calculate the inverse of $M$ such that: $M \in M_{2n}(\mathbb{C})$ and $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$, and I find that $\det M = 2^n$. I tried to find the ...
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2answers
79 views

Why is $3^n$ not in $\Theta(2^n)$

How is it that $3^n$ not in $\Theta(2^n)$, while $log_3 n$ is in $\Theta(log_2 n)$ ?
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64 views

Pseudo-inverse of an underdetermined Toeplitz matrix

I have an undetermined Toeplitz matrix (more columns than rows). For example: \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 ...
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2answers
38 views

how to find inverse laplace transform of

how to find the inverse laplace transform of $\frac{s}{s^4+s^2+1}$. I tried to do it via partial fraction and reached $\frac{s}{(s^2-s+1)(s^2+s+1)}$
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2answers
75 views

Inverse of $f(x) = 18sin(\frac{x\pi}{7})+20$

This is an exercise taken from Mooculus-textbook (page 17, exercise 5 to be exact). The task given is to find an inverse for $f(x) = 18\sin(\frac{x\pi}{7})+20$ (restricting domain to $[3.5,10.5]$) ...
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1k views

calculator issue: radians or degrees for inverse trig

It's a simple question but I am a little confused. The value of $cos^{-1} (-0.5)$ , is it 2.0943 or 120 ?
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3answers
159 views

Do continuous mappings always have an inverse?

A theorem of general topology states that: A mapping $f$ from $X$ to $Y$ is continuous if and only if the inverse image of any open set in $Y$ is open in $X$. Does this mean that continuous ...
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2answers
144 views

Does a smooth mapping always have an inverse map which is also smooth?

If not, can someone provide counterexamples? Thank you
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1answer
52 views

Optimal series expansion for “invertability”

Motivation: Often when dealing with physical phenomena, deviations from the model must be considered, so a variable, say $x\in[0,1]$ will be replaced by a power series expansion: $$x'\ \to \ x(1+k ...
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3answers
136 views

Invertible function $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $

How can I prove that $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $ is invertible. First I choose variable $x$ for $y$ and tried to switch and simplified the function but I am stuck. Need some help ...
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46 views

Integrating inverse functions

I'm trying to integrate the following: $$\int_0^1 \left[\frac{c}{(1+c^{-1}(\tilde{b}))}\right]dc$$ If it helps ...
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1answer
38 views

Invertibility of a Matrix Given Some Conditions

Let $A$ and $B$ be different $n\times n$ matrices with real entries. Suppose that $A^3=B^3$ and $A^2B=B^2A$, can $A^2+B^2$ be invertible?
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1answer
71 views

Inverse Z transformation in specific points

I'm given $$H(z)=\frac{z^4+6z}{z^6+1}$$ and I need to find $h(k)$ for $k=0,1,2,3,4$. Where $H(z)$ is the Z transformation of $h(k)$. Since H is very complicated, I believe some trick could be ...
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2answers
31 views

Matrix inversion with variable in {-1,1}

Could you please give me a hint for computing inversion of this matrix? $$ \begin{pmatrix} 1 & f & g+h\sqrt(2) \\ 0 & i & j \\ 0 & 0 & 1 \\ ...
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1answer
567 views

Inverse of binary matrix

I have tried creating an inverse of a binary matrix using the identity matrix method. Have an identity matrix alongside the square matrix and perform all the operations to convert the square matrix to ...
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1answer
43 views

Solving: How to find an inverse function for this function?

I got this example: and I am trying to find an inverse function to this function. Could I ask you, please, how to do that? Thank you
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1answer
224 views

Reversion of power series

So, I just heard about this method. How does one determine the coefficients, and what is it used for? For example, given $$ y = x - \frac{x^3}{6} + \frac{x^5}{120} + O(x^7)$$ reversion would give a ...