Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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Derivative of Nested Matrix Quadratic Form

I have two real matrices: $\mathbf{A} \in \mathbb{R}^{k \times d}$, $\mathbf{B} \in \mathbb{R}^{d \times d}$, where $k \leq d$. Further $\mathbf{B}$ is symmetric. I also have two vectors $\mathbf{c},\...
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17 views

Matlab inverse tangent

I have a system that contain complex conjugate pole pair in his transfer function. If I examine the function with Matlab's bode it works great and I get results I ...
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1answer
43 views

Find the original function by using convolution theorem

Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it. \begin{equation}\frac{1}{((...
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52 views

L2 Norm of Inverse of Non-square Matrix Multiplication

Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$. Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/upper bound for $\|(AA^T)^{-1}\|_2$ in terms of $\...
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Finding the multiplicative inverses in fields

Let's say I have the field $F_{11}$. Why does $2$ have the multiplicative inverse $6$? In some of the examples I have, let's say we are looking $F_5$, why are values up to only $2$ considered? So ...
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Finding a matrix inverse when an equation involving it is a multiple of the identity matrix

Say you had a matrix $A$, and you did an equation like $A^2 - A$, and proved that it was a multiple of $I$. How could you find $A^{-1}$ in the form $rA + sI$ after proving that? I want to do it ...
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424 views

Let A be a square matrix such that $A^3 = 2I$

Let $A$ be a square matrix such that $A^3 = 2I$ i) Prove that $A - I$ is invertible and find its inverse ii) Prove that $A + 2I$ is invertible and find its inverse iii) Using (i) and (ii) or ...
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1answer
41 views

What is meant by In-Place Matrix Inversion?

I come across the term "In Place Matrix Inversion" a lot in numerical libraries like NumPy and ND4J. What does it mean ? How is it different from the normal matrix inversion ? What are the advantages ...
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30 views

Matrix Inverse as Series

I am looking for different representations of the inverse of a matrix as a power series. One obvious candidate is the Von Neumann series which is given $$A^{-1} = \sum_{k=0}^{\infty} (I-A)^k$$ ...
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61 views

Solve equation of inverse functions

I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and $f^{-1}_2(x)$, by ...
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328 views

Definition of Inverse in Linear and Abstract Algebra

In a linear algebra text, the following is the definition of the inverse of a matrix An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = I_n$$...
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77 views

In which cases are $(f\circ g)(x) = (g\circ f)(x)$?

I have found three cases: 1) If $f$ and $g$ are the same function. 2) If $f$ and $g$ are mutually inverse. 3) If both are polynomials of degree $1$ Maybe there are more.
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51 views

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
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2answers
55 views

Find the inverse $\dfrac{x}{\|x\|}$ in $\mathbb{R^2}$

I wish to find the inverse of $\dfrac{x}{\|x\|}$, where $x \in \mathbb{R}^2$ Let's do this. Let $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}}$$ $$y_2 = \dfrac{x_2}{\sqrt{x_1^2+x_2^2}}$$ Then $$y_1 = \...
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32 views

How to simplify inverse trigonometric function

How to simplify the following equation: $$\sin(2\arccos(x))$$ I am thinking about: $$\arccos(x) = t$$ Then we have: $$\sin(2t) = 2\sin(t)\cos(t)$$ But then how to proceed?
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1answer
19 views

Generate random variate using inverse transform technique of $ f (x) =a (1+|x-2|)$

I need to generate a random variable with density function: $$ f(x)= \begin{cases} a (1+|x-2|) , & {-1 \le x \le 4} \\ 0, & \text{elsewhere} \end{cases} $$ For that I need to inverse the ...
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Why does $\frac{1}{6e^{2y}}=\frac{1}{2x-8}$ in this context?

This is the context: I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
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Asymptotes of $\arctan (2x)$

My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, ...
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1answer
19 views

Geometric progression with reverse order

I have the following problem: Find three positive numbers which have the sum of $70$ and create a Geometric progression ($q>0$, increasing). Their inverse sum equals to $4/70$. Thank you!
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54 views

Is a factorable polynomial invertible?

The reason there exists no quintic formula that finds the roots of a quintic polynomial is simply because some quintic polynomials are irreducible. But reducible quintic polynomials may be invertible ...
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58 views

Can you inverse a funcion by rotating it?

In school i sometimes run on some excercises where you need to calculate something that has an inverse function in it but you cannot find the inverse and you need to work your way around it. I know ...
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1answer
41 views

Inverse of the composition of two functions

If I have a composition of two functions: $$y = f(g(x),h(x))$$ where both $g(x)$ and $h(x)$ are readily invertible, can I find the inverse of the composition? i.e.: Can I find $x = f^{-1}(y)$? I ...
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1answer
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Invariant under $x \rightarrow 1/x$?

I started thinking on the following problem. I am interested in finding complex functions of a complex variable such that $\phi(z)=\phi(z^{-1})$ So far, all I could come up with was a family of ...
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31 views

Does $\sin^{-1}x$ has a vertical tangent

I read that the function $f(x)$ has a vertical tangent at $x=a$ in the domain of $f$ if $$f'(a^-) \to +\infty$$ and $$f'(a^+) \to +\infty$$ Or both approach to $-\infty$. But for $f(x)=\sin^{-1}x$ $...
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55 views

For which values of $a,b$ is the matrix invertible?

I am trying to figure out the below question: 15. For which values of the constants $a$ and $b$ is the matrix $$A = \left[\begin{array}{cc} a & -b \\ b & a \end{array}\right]$$ ...
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20 views

Integral inversion

Say I know this function $$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$$ where $m(x)$ is a Fourier transform of an infinitely differentiable real function, whose maximal ...
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86 views

Can the cross product of two non-invertible matrices be invertible?

To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible? Just used to help understand a Linear Transformation assignment question, don't ...
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72 views

condition number of matrix plus constant times identity

I saw this post on the eigenvalues of a matrix plus a constant times the identity matrix. Say $A$ is an $n\times n$ matrix (real and non-singular) with eigenvalues $\lambda_1,\ldots,\lambda_n$, then ...
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32 views

Inverse image of the set $[−1, 4)$ under $f : x \mapsto -x^2$

I have an issue with determining the inverse image of a set. I cannot understand anything about it. I've got a simple exercise here, could someone here show me how the inverse image works and more ...
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1answer
37 views

What is $\ln(e^x -4) $, solving for the inverse?

What is $\ln(e^x -4) $, solving for the inverse? I know $\ln(e^x)$ is just $x$, but I don't know what to do with the 4.
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Finding Inverse of a matrix using elementary transformations

So I have to find the Inverse of A. $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 3 & 4 & 3 \\ \end{bmatrix} $$ By using elementary row or column transformations.. The ...
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44 views

Numerical Algorithm for $n \times n$ Matrix Inverse

I have to write a C program in which I have to compute the matrix inverse of a $n \times n$ matrix. Is there a convenient iterative process that I can use to do that? All I see is the co factor method ...
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1answer
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Special Case Linear Solvers

I, and friends of mine, are interested in matrices which can be inverted / solved easily (i.e. in less than O(n^3)). I started to put together a github page dedicated to it and so far have identified: ...
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10 views

Invariance of Frobenious norm under transformation.

Can we say for every invertible square matrix $\mathbf{P}$, $\Vert\mathbf{X-B}\Vert_F^2=\Vert\mathbf{P^{-1}(X-B)}\Vert_F^2$. And does this hold true for non-square matrix $\mathbf{P}$ under some ...
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76 views

How does $1 + \tan^2x = 1/\cos^2x$?

I am unable to see why $$1 + \tan^2 x= 1/\cos^2x$$ I have looked into the topic anad I am familiar with the reciprocal ratios of cosec, sec, and cot. but cannot derive how this statement makes sense. ...
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1answer
22 views

Having a holomorphic function $h$ that's the inverse of a function $f$, it's also the inverse for a continuation of $f$

Let $\gamma: [0, 1] \to \mathbb{C}$ be a (continuous) path, $\gamma(0) \in D$, $(f, D)$ a tuple of a holomorphic function $f: D \to \mathbb{C}, D \subseteq \mathbb{C}$ a simply connected open set. Let ...
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46 views

Proving matrix properties: [closed]

Prove: (i) $A(I+BA)^{-1}=(I+AB)^{-1}A$ (ii) $(I+AB)^{-1}=I-A(I+BA)^{-1}B$ (i) Consider $A(I+BA)=(A+ABA)=(I+AB)A$ Taking inverse on both sides (invert) $[A(I+BA)]^{-1}=[(I+AB)A]^{-1}$ $...
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1answer
48 views

Using Chinese Remainder Theorem to find an integer $x$ for which $ x\equiv 3\pmod 4 x\equiv 5\pmod 9 x\equiv 10\pmod {35} $

Hello I have got problems with understanding the reduction method in CRT. We have got system like this $$x\equiv 3\pmod 4$$ $$x\equiv 5\pmod 9$$ $$x\equiv 10\pmod {35}$$ There is a way to do this ...
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1answer
62 views

Inverse of matrix with particular structure

I have a square invertible matrix $A=[c, c^2, c^3 \dots c^n]$ where $c \in \Bbb R^n$. Are there any known fast tricks for inverting it? Edit: $c$ is a column vector and raising it to a power is to ...
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48 views

What approximations for the Gamma function's inverse appear to work 'best'?

So I was wondering how we approximate the inverse of the Gamma function, where I tried a few methods: Lagrange inversion theorem: $$\Gamma^{-1}(z)=a+\sum_{n=1}^{\infty}\lim_{w\to a}\frac{(z-\Gamma(a)...
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Inverse trigonometric expansion related question

I know expansions for $\sin^{-1}(x)+\sin^{-1}(y)$, but does there exists any expansion for $\sin^{-1}(x \pm y)$ if not then what is the reason?
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How do I prove that $(ABC)^{-1} = C^{-1} B^{-1} A^{-1}$ [closed]

Please help me answering this problem! thank you :) Prove that for any nonsingular matrices $A$, $B$, and $C$, the equation $$(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$$ holds. (Hint: Assume $D$ is the ...
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what condition of A makes transpose(A)*A nonsingular?

What contidion of A makes $$A^TA$$ nonsingular? If so, that is $$A^TA$$ is non-singular than a unique solution exists.
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partially ordered group, does x=-x imply x=0?

I have just a simple question: Let (G,+) be a partially ordered Abelian group. Does x = -x imply x = 0 ? If the answer is yes, then how could i prove it? If the answer is no, then a ...
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1answer
31 views

Find $\sinh^{-1}x$

The hyperbolic sine function, $\sinh(x)$ , is defined by the equation: $$ \sinh(x) = \frac {e^x-e^{-x}} {2}$$ Find a formula for its inverse, $$ \sinh^{-1}(x) $$
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1answer
33 views

Why arsin function has range $[-\pi/2,\pi/2]$ [duplicate]

While studying in P.75 of inverse trigonometric functions it tells we have to restrict our domain before finding the inverse.But I can't get why we choose $[-\pi/2,\pi/2]$?Why can't we choose $[\pi/2,...
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13 views

Ensure that for each number in specific space there is inverse

Let say I want to find the Inverse number of some serial number. ( 9 digits number .. its can be an ID). And let say we want to find the inverse in $\mathbb Z_{1000000123}$ ( for example ) How I can ...
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125 views

How to convert $(A+\lambda E)^{-1}$?

Here is one of the most famous equation called Sherman–Morrison formula (1951) when we want to get an inverse matrix. $$(A+vw^{\text{T}})^{-1}=A^{-1}-\cfrac{A^{-1}vw^{\text{T}}A^{-1}}{1+{w}^{\text{T}}...
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69 views

Getting $B$ from $A = M^t B M$ without inverting $M$

I have got three matrices: $A$ (dimension $n \times n$), $B$ (dimension $m \times m$) and $M$ (dimension $m \times n$). We have $m > n$. This is the relation between these three matrices: $A = M^t ...
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153 views

How to invert this expression involving $\tanh^{-1}$?

I've got the expression: $ x = \tanh^{-1}(p) - \sqrt{\frac{2}{3}} \tanh^{-1}\left( \sqrt{\frac{2}{3}} p\right) $ How can I invert this function so I have a function $p(x)$? I thought about using ...