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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What would be the inverse function for the following condition?

What would be the inverse function condition for the above question.
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Determining $f^{-1}(3)$ without knowing $f^{-1}(x)$ but given $f(1)=3$ and $f'(x)>0$.

I have a continuous function $f(x)$ and I want to find $f^{-1}(3)$, but I can't find $f^{-1}$ directly. I know that $f(1)=3$ and $f'(x)>0$ for all x. Because the function is continuous and always ...
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61 views

Inverse of $x^2+\log^2\cos x$

I'm looking for the inverse of $$f(x)=x^2+(\log\cos x)^2$$ Where $f$ is defined from $[0,\pi/2)$ It dosen't have to be closed form, a sum, an integral or some special functions would be of interest ...
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2answers
23 views

Prove that $B$ is invertible:$B = A_{11} - A_{12}A_{22}^{-1}A_{21}$ if…

Let $$A = \begin{bmatrix}A_{11}&A_{12} \\ A_{21}&A_{22} \\ \end{bmatrix}$$ Prove that $B$ is invertible:$$B = A_{11} - A_{12}A_{22}^{-1}A_{21}$$ $$$$ in case of $A$ and $A_{22}$ being ...
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Formal inverse of a matrix ressembling Fourier's matrix

What is the formal inverse of a square $N\times N$ matrix $A$ with entries $A_{ij}=a^{(i-1)(j-1)}$? When $a$ is the $N$th root of unity (i.e. $a=\exp(2 \pi i/N)$), then $A$ is the Fourier matrix and ...
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1answer
21 views

Complex Field - Proving $\left(\frac{z_1}{z_2}\right)^{\star} = \left(\frac{z_1^{\star}}{z_2^{\star}}\right)$

Like the title states, I'm trying to prove that $\left(\frac{z_1}{z_2}\right)^{\star} = \left(\frac{z_1^{\star}}{z_2^{\star}}\right)$ where z is a complex number and z* is its conjugate. I keep ...
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1answer
69 views

Is there a closed form for the inverse of $y=x^{x^x}$?

It's pretty well known, and easy to derive, that $y=x^x$ has the inverse $y=\frac{\ln x}{W(\ln x)}$. I've had no luck trying to work out the inverse of any larger power towers, though. Is there any ...
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1answer
39 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
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202 views

How do I solve $x^5 +x^3+x = y$ for $x$?

I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$? The explicit question out of the ...
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1answer
83 views

How to find $10^{-1}$ in $\{0, 2, 4, 6, 8, 10, 12\} \subseteq \mathbb Z_{14}$?

Here is my question: Consider the subset $S = \{0, 2, 4, 6, 8, 10, 12\}$ in $\mathbb Z_{14}$, with the operations of addition and multiplication in $\mathbb Z_{14}$. (a) Show that $S$ has ...
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1answer
113 views

Determinants of Matrices det(4A) equals?

Suppose A is a 4 x 4 matrix such that det(A) = 1/64. What will det(4A^-1)^T be equal to? Here's my thinking, det(A^T) = det(A) I has no effect on the determinant. And det(A^-1) = 1/det(A) so ...
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0answers
38 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
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1answer
22 views

Solving least-squares: why ever use iterative descent methods over pseudoinverse?

I recall doing an assignment in machine learning where we ran regression tests on a data set, both using our own implemented gradient descent program, and then using the (right) pseudoinverse ...
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17 views

The inverse of a Moment generating function

The moment generating function of $X$ is $M_X(t) = \mathbb{E}[e^{tX}] = \int e^{tu}f_X(u)du$ where t is a complex variable and $f_X$ is the density of X. The cumulant generating funtion of $X$ is ...
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2answers
65 views

Evaluating an inverse function by sketching a unit circle

Problem I'm working on: "Evaluate the inverse function by sketching a unit circle, locating the correct angle and evaluate the ordered pair on the circle." The function I got was $\cos^{-1}(0)$. ...
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32 views

Local inversion theorem (théorème d'inversion local)

I don't understand how to use the local inversion theorem to prove that a nondegerate critical point of a function $f\in C^2(U,\mathbb{R})$ is isolated Thank you.
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2answers
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Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$ [closed]

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots ...
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2answers
38 views

Inverse of $x(x+2)$ given $x\ge -1$

Consider the function: $y=x(x+2)$ . Consider its domain to be $x \geq -1$ . Graphically it makes sense that the inverse of this function is $-1 + \sqrt{x+1}$. But how to compute it analytically? ...
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39 views

How to show the surjectivity of $f(x)=x^5$ on $\mathbb R$?

Sasy $f:\mathbb R\to\mathbb R$ define by $f(x)=x^5$ This is definitely injective as $x_1^5=x_2^5 \implies x_1=x_2$ I say it is surjective because for all really $x$ there is all real $y$, $x \in ...
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4answers
41 views

How to find a modular multiplicative inverse when GCD is not 1

I am working on a problem that requires finding a multiplicative inverse of two numbers, but my algorithm is failing for a very simple reason: the GCD of the two numbers isn't 1. I figured I must've ...
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29 views

Looking for reference for the criterion of inveribility of a difference of two invertible matrices

It is pretty easy to show that $A-B$ is invertible if either $AB^{-1}$ or $BA^{-1}$ have all eigenvalues of absolute value less than $1$. But I am specifically looking for a handy reference of this ...
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If a one-to-one function's inverse is the same what must be true of the graph of f?

As a followup to this question. I'm trying to determine what must be true of the graph of $f$ in these cases. I've examined the two functions $f(x)= x$ and $f(x)= \frac{1}{x}$ and I'm not seeing any ...
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1answer
71 views

Can an Elementary Matrix's Inverse's Determinant = 0?

Can someone explain to me why an elementary matrix's inverse determinant cannot equal 0? Or can it? Is there some theorem to elementary matrix inverses? THANKS FOR YOUR INSIGHT! :)
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1answer
42 views

Finding the area bounded by two curves when in terms of $x = y^2$?

I can't seem to figure this problem out. Find the area bounded by the curves $x=2y-y^2$ and $x=4-y^2$, in the first quadrant. I am having difficulties with graphing the equations and coming up ...
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1answer
50 views

Solving a set of non-linear matrix equations

Consider the following set of equations $$\begin{cases}PAQ^{-1}&=T \\ QBR^{-1}&=T\\ RCP^{-1}&=T, \end{cases} $$ where A,B,C and T are known real-valued $3\times3$ matrices and P, Q, R are ...
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1answer
20 views

Show that a square matrix with linear transformation T(M) = A·M is bijective when A is inversible

Suppose $K$ is a body (a field), $n ≥ 1$ and $A ∈ M_n(K)$ a fixed matrix. Consider the linear transformation $T : M_n(K) → M_n(K)$ defined by $T(M) = A · M$ for $M ∈ M_n(K)$ The mark scheme says ...
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61 views

Why does the square root of an inverse function turn negative?

For example, $$f(x)=x^2$$ $$y=x^2$$ $$-\sqrt{x} = f^{-1}$$ Why does $\sqrt{x}$ become negative? Edit: Sorry for all the confusion, I will state the problem on my textbook and the solution. ...
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151 views

Determine the greatest interval where the function is invertible

The assingment is to determine the greatest interval around $x=0$ where the function: $$f(x)=x^5-5x+3$$ is invertible. After that, determine $(f^{-1})'(3)$ I have totally forgotten all about ...
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1answer
38 views

Proving that there is no invertible matrix with zero row sums using determinants

I have the following question which I know I should use the determinant to solve. Here it is: Determine if there exists an invertible $3\times3$ matrix $A$ such that $$\begin{align*} ...
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4answers
74 views

If a function $f$ is decreasing on its domain then would its inverse be increasing or decreasing?

I have a question concerned the inverse of a function $f$ and the sign of its derivative. If we are given a function $f$ that is decreasing on its domain, would its inverse $f^{-1}$ be increasing or ...
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1answer
40 views

Existence of solution for matrix equation $ (I - \alpha A) \bar{x}=\bar{b}$

This is my first question in here and I would be really thankful if someone could help me with understanding the matter. I am solving a matrix equation $(I-\alpha A) \bar{x} = \bar{b}$ for a positive ...
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1answer
29 views

Finding an inverse laplace transform for $\displaystyle\frac{a}{\left(s^2 + a^2\right)^2}$

I am asked to show that $x'' + w^2x = f\sin(wt)$ has a solution given by $x = \frac{f}{2w^2}(\sin(wt) - wt\cos(wt))$ where $w$ and $f$ are constants, by means of Laplace transforms. By taking a ...
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30 views

How to show that a null potent linear transformation is invertible

V is a K vector space and $ψ : V → V$ is a null potent linear transformation i.e. $ψ^N = 0$ for a certain $N ∈ N$. Prove that $Id_V − ψ$ est an invertible element in the ring $L(V, V )$. My assistant ...
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61 views

Prove that $(a+b\sqrt[3]{2}+c\sqrt[3]{4})^{-1}$ with a,b,c∈Q is a number of the form $d+e\sqrt[3]{2}+f\sqrt[3]{4}$ with $d,e,f∈Q$ [duplicate]

Prove that $(a+b\sqrt[3]{2}+c\sqrt[3]{4})^{-1}$ with $a,b,c∈Q$ is a number of the form $d+e\sqrt[3]{2}+f\sqrt[3]{4}$ with $d,e,f \in Q$ I'd like to do this without using too much fancy ...
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3answers
118 views

How to find the inverse function in explicit form?

For a function below: $$f(x)=a\cdot e^{-k_1 x}+b\cdot e^{-k_2 x}$$ How can I obtain its inverse function in explicit form?
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1answer
59 views

Proof of the rank theorem in Rudin's PMA book

I am studying Rudin's proof of the rank theorem (theorem 9.32 in Principles of Mathematical Analysis.) We have an invertible function $H(x)$ defined on an open set. He claims we can "shrink" the open ...
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1answer
29 views

Pseudoinverse and orthogonal projection

Given the matrix $A= \begin {pmatrix} 1 & 1 &1 \\ -1 & 1 & 0 \\ 0 & 2 &1 \end{pmatrix}$. (i) Determine the orthogonal projection $p:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ on ...
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1answer
45 views

Help to prove the existance of a function

Let $f:X \rightarrow Y$ be a function. Prove that there exists a function $g:Y \rightarrow X$ such that $f \circ g = I_Y$ if and only if $f$ is a surjection. I need help on proving the following: ...
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0answers
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Which (approximative) methods are there to compute the inverse of a complicated function?

I have a complicated function $f(x)$ for which I want to compute the inverse $f^{-1}$ over a certain range $R(f): a \leq f(x) \leq b$. The only way to find the inverse I can think of is power series ...
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3answers
39 views

inverse of quadratic log functions

Can a Log function with a quadratic have an inverse function? The specific question is to find the inverse of $$f(x) = \log_2(x^2-3x-4)$$ The function already fails the horizontal line test, but ...
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1answer
66 views

Power series and their inverses (radius of convergence of each)

Suppose I have a power series approximation $y$ to an invertible function $f(x)$, and I know that $y$ convergences around $x$ on an interval $(-R,R)$, $R$ being the radius of convergence. How are the ...
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0answers
52 views

Find a multiplicative inverse of an element in a field

Suppose we have an element $\sigma=p+qa\rho+rd\rho^{-1}\in K$ where $K=\mathbb{Q}(\rho)$ where $[K:\mathbb{Q}]=3$ I want to find a multiplicative inverse of $\sigma$ i .e ...
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0answers
26 views

Can anyone give the equation of the inverse of radial projection from a tetrahedron to sphere?

$(x,y,z) \mapsto \bigg(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}} \bigg)$ This is the equation of the radial projection. I need the inverse of this ...
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1answer
31 views

Express summation in terms of matrix norm

Express the following $$\sum _{ i=1 }^{ n }{ ({ \beta }_{ 1 }x_{ i }+{ \beta }_{ 0 }-y_{ i })^{ 2 } }$$ To become something of the form: $∥Ax−b∥^{ 2 }$ where $A$ is an $m$−by−$n$ matrix and $b$ is ...
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1answer
53 views

Fast way to inverse B'CB+D

$\mathbf {A = B'CB}$, where $\mathbf A$ is of dimension $n \times n$, $\mathbf C$ is m by m, positive definite and symmetric, $\mathbf B$ is of dimension $m \times n$, and $n >> m$. Inversion ...
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1answer
116 views

Normalization of a two-dimensional kernel function

I've got three two-dimensional kernel functions which look like this $$ k(r,h) = n \cdot \begin{cases} \ldots & 0 \le r \le h \\ 0 & otherwise \end{cases} $$ With ...
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1answer
73 views

It $f(x)=x+\sin x$, then can we find $f^{-1} (x)$?

We have a bijective function $f(x)=x+\sin x$. So what is $f^{-1} (x)$? Let $f^{-1}(x)$ be $g(x)$. Suppose we have to find $g\left(\dfrac{\pi}{6}+\dfrac{1}{2}\right)$ and ...
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0answers
59 views

How to find the inverse function of f(x)=x+sin(x)-a

The problem is how to find the inverse function of $$f(x)=x+\sin(x)-a$$ where $a$ is real parameter. I tried to write $\sin(x)$ as $\frac{i}{2}(e^{-ix}-e^{ix})$. Problem is how to solve this equation: ...
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2answers
51 views

Inverse of 2d function involving sine and cosine

I have the function $f: \mathbb R^2 \to \mathbb R^2$ or more precisely $$f\left([0,\pi/2]^2\right)=\{(x,y) \in \mathbb R^2 : \Vert (x,y) \Vert \leq 1 \text{ and } y\geq0\}$$ which means it is a ...
0
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1answer
22 views

Find the point of $f^{-1}$ corresponding to the value of x indicated

i am having problems understanding this problem. The given function $f$ is one-to-one. Find $f^{-1}$, find the point on the graph of $f^{-1}$ corresponding to the indicated value of $x$ in the ...