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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solution of matrix equation

I was trying to solve the problem I have posted previously (here). and stuck up at the point where I need to find a simplified expression for $(\mathbf{I-DW})^{-1}$ Where $\mathbf{W}$ is a doubly ...
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3answers
47 views

If a function maps an input to its inverse, is it bijective?

I read in my textbook that a function is a bijection if and only if it has an inverse. Is it the same thing to say a function $f: X → X$ is a bijection if $f(x) = x^{-1}$? If $a = x$ and $b = x^{-1}$, ...
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1answer
17 views

Knwing when the inverting operation were wrong with $A^{-1}A$ result

I don't know why but I'm really really weak in inverting matrices since years... I always do some mistakes. I'm asking you how could I cope with that problem and be able to invert matrix easily in the ...
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0answers
24 views

differentiation of a norm of matrix function

I need to differentiate the following function W.r.to $x$ $y=\|x (\mathbf{I-W}-x \mathbf{Diag(v_2)W})^{-1}\mathbf{v_1} - b\|_2$ where $0<x<\frac{2}{max_i{|{v_2}_i|}}$,$\mathbf{v_1}\in ...
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2answers
740 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 ...
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2answers
34 views

AP Calculus BC - Derivative of inverse problem

Let $g(x)$ be the inverse of the function $f(x)$. Given the following values on the table below, at which value $x=a$ will $g'(a)=1/6$? (No calculator allowed) ...
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2answers
74 views

Finding the inverse of $f(x) = x^3 + x$

How can one find the inverse of such functios?. I know how to do it for explicit quadratic functions; how do I express $x$ as a function of $y$ here?
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2answers
116 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 ...
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19 views

Inverse of a function and Inverse function theorem.

Apologies if this question is too primitive for professionals here. I understand the inverse of a function, in terms of domain, co-domain and bijections. Let's say $f,g:[0,\infty)\rightarrow ...
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1answer
41 views

Inverse of a “Vandermonde-like” matrix composed of power series

Is there an analytical formula for the inverse of a complex matrix whose elements are sets of "power series" except the last term is scaled? Let $0<x_1<x_2<...<x_n$ be monotonically ...
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1answer
17 views

When diagonalizing a matrix, in what order should you arrange the the eigenvectors to form the invertible matrix $P$?

I was following this example online to diagonalize a matrix. It lists the eigenvectors as $\lambda =3,2,4$ (note the order). It then arranges each eigenvalue's corresponding eigenvector (3 column ...
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2answers
75 views

When does $A^TAX = A^TB$?

The original question is a true-or-false question: Assume $A$ is a $m\times n$ matrix and $B$ is a $m\times p$ matrix. If $X$ is an $n\times p$ unkwown matrix, then the system $A^TAX = A^TB$ always ...
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0answers
11 views

how to calculate cumulative distribution function inverted exponential in this pic

how to calculate cumulative distribution function inverted exponential in this pic enter image description here
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1answer
1k views

Prove that a matrix with a trivial nullspace must always be invertible

I'm proving that $A^tA$ will be positive definite iff $A$ is invertible. I guess that there are ways to show this with determinants, eigenvectors. But I've just gone with Positive definites must ...
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2answers
35 views

Inverse of a square block matrix

I am trying to understand how to compute the inverse of a square block matrix defined as follow: $\begin{bmatrix}2{\bf I}&-{\bf X}\\{\bf X}'&{\bf 0}\end{bmatrix}$, where ${\bf I}$ is a ...
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2answers
478 views

How would I show this bijection and also calculate its inverse of the function?

I want to show that $f(x)$ is bijective and calculate it's inverse. Let $$f : \mathbf{R} \to \mathbf{R} $$ be defined by $f (x) = \frac{3x}{5} + 7$ I understand that a bijection must be ...
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2answers
68 views

Prove that if $\|A\|<1$, then $(I+A)^{-1}=I-A+A^2-A^3+\cdots$. [on hold]

Prove that if $\|A\|<1$, then $(I+A)^{-1}=I-A+A^2-A^3+\cdots$. I'm not sure how to do this. I know the result for $(I-A)^{-1}$, but that won't help me.
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2answers
20 views

Inverse of a matrix with uniform off diagonals

Suppose that we have an all positive matrix where the off diagonal elements are all identical. Can one calculate the inverse of the matrix analytically, or more efficiently than the general case? For ...
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3answers
69 views

If $f$ function then $f^{-1}$ function iff $f$ function injective (one-to-one).

During the lecture we learned this phrase: "If $f$ is a function then $f^{-1}$ is a function iff $f$ is injective (one-to-one)." But why? What with onto? $f$ doesn't need to be Surjective ...
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5answers
77 views

Solving $\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$

If we have $$\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$$ we have to find the set of $x$ for which this is true. I tried to solve it by putting $x = \sin a$ or $\cos a, but got no ...
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1answer
2k views

Find an inverse of $a$ modulo $m$ for each of these pairs of relatively prime integers

How would I find the inverse of a given number $a$ modulo $m$, given that $\gcd(a,m)=1$? a) $a = 2$, $m = 17$ $17 = 2 \cdot 8 + 1$ $2 = 1 \cdot 2 + 0$ $1 = 17 - 8 \cdot 2$ <-How do I know ...
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2answers
51 views

Solving $\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$

If we have to find the solutions of equation $$\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$$ Using a triangle I rewrite it as $$2 \arctan ...
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38 views

Let A=$\tiny\begin{pmatrix}1&1&1\\1&2&2\\ 1 & 2 &3 \end{pmatrix}$ and B=$\tiny\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 &0 \\ 1 & 1 &1 \end{pmatrix}$

Then (A) there exists a matrix C such that A = BC = CB (B) there is no matrix C such that A = BC (C) there exists a matrix C such that A = BC, but A $\neq$ CB (D) there is no matrix C such that A ...
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1answer
23 views

Laplace transform and inverse laplace transform

1- Find laplace transform for $4e^2t-3\cos^2(2t)+2\cosh(3t)$ My answer $L(4e^2t-3cos^2(2t)+2cosh(3t))=4L(e^2t)-3L(cos^2(2t))+2L(cosh(3t))$ $=\frac4 {s-2}-3L(\cos^2(2t))+\frac{2s}{s^2-9}$ But how ...
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1answer
28 views

Verifying multiplicative inverses of modulo n are the elements that are relatively prime to n

A proposition in my book states: $(\mathbb{Z}/n\mathbb{Z})^{\times} = \{a \in \mathbb{Z}/n\mathbb{Z}~|(a,n) = 1\}$ which I want to prove. I start by defining $a$ in terms of prime factors $$a = ...
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1answer
18 views

symetric matrix inverse

Is there an easy way to invert a 3x3 symmetric matrix? for example A = $\begin{pmatrix} -1& 2& 0\\ 2& -5& 0\\ 0& 0& ...
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1answer
12 views

Find the inverse of $θ:P(\Bbb{Z})→P(\Bbb{Z})$ defined as $θ(X) = \bar X$

Find the inverse of $θ:P(\Bbb{Z})→P(\Bbb{Z})$ defined as $θ(X) = \bar X$ (the complement of $X$)? Would the inverse of the function just be the function itself?
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22 views

Changed codomain of inverse trigonometric functions

If codomain of $\arcsin(x)$ is $(\pi/2 , 3\pi/2)$ and codomain of $\arccos(x)$ is $(\pi , 2\pi)$ then what should be $\arcsin + \arccos$ equal to ? I thought of putting $x = \sin \theta$ But then ...
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0answers
20 views

inverse eigenvalue problem [closed]

algorithm for extended hessenberg inverse eigenvalue problem in matlab? is there any program codes in matlab for solving this mathematical problem expressed above?
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1answer
69 views

Prove $sgn(π) = sgn(π^{-1})$?

I'm pretty sure the inversion count of $π$ should be the opposite of the inversion count of $π^{-1}$. By this I mean if $π$ looks like this: $1 \to 1$, $2\to 2, \ldots, 10 \to 10$ and therefore the ...
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12answers
3k views

Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
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1answer
48 views

Inverse function to $f(t)=3t+4ln(t+1)=y$

I have to invert the function $f(t)=3t+4\ln(t+1)=y$, so $f^{-1}(y)=t$. But I am struggling to invert this. Is there a solution?
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1answer
486 views

How to find the Frechet derivative at $A\rightarrow A^{-1}$ mapping?

I am reading on my own the Lectures on the Geometry of Manifolds (http://nd.edu/~lnicolae/Lectures.pdf ) , and got stuck in solving the exercise 1.1.3 (b) . The 1.1.3 (b) is : Let F: $U\rightarrow ...
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2answers
21 views

Simplifying Inverse Trig Function

I'm trying to figure out how to simplify this expression but I'm not quite sure on how to approach this question. How should I approach this question? Any help is greatly appreciated! ...
3
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1answer
26 views

Show that a matrix satisfying certain conditions is non-singular

I have a square matrix $A$ satisfying the following conditions: The elements on the diagonal are negative; All other elements are non-negative; All row sums are less than or equal to $0$; There is ...
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1answer
24 views

I need someone to show me how to solve this input/output problem

Alright, so I have: $4y^3 = x$ And now I have to solve for $y$, where I can later use that equation to answer other questions I have. Can someone hint me out on how to solve for $y$ given the above ...
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1answer
20 views

Inverse of the sum of a invertible matrix with known Cholesky-decomposion and diagonal matrix

I want to ask a question about invertible matrix. Suppose there is a $n\times n$ symmetric and invertible matrix $M$, and we know its Cholesky decomposion as $M=LL'$. Then do we have an efficient way ...
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0answers
15 views

The existance of Schur Complement Inverse

A block matrix $\mathbf{M}=\left[ \begin{array}{ccc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^T & \mathbf{C} \end{array} \right]$ is invertible if $\mathbf{A}$ and ...
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3answers
38 views

Is the inverse of a real, continuous “1-1” function necessarily continuous itself? [closed]

If so, please do provide me with an epsilon-delta proof, if possible. Thanks in advance.
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21 views

Using Euclid's algorithm, how do I find a polynomial $f_p(x)$ such that $f \cdot f_p \equiv 1 \pmod p$?

Suppose that we have a polynomial $f(x)$ with coefficients in $\mathbb{Z_3}$ and maximum degree $N-1$, where $N$ is prime. (In fact, we consider $f$ as a class of ...
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1answer
20 views

Related to symmetric, diagonal and invertible matrices

While solving a problem I came across a specific question: Given $A,B$ as $2$ real, symmetric, matrices with $B$ positive definite, does there exist a matrix (invertible) $P$ such that both $P^TAP$ ...
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12 views

Harmonic inversion of an eccentric circle.

Inverted here is a circle with respect to another circle not as the conventional reciprocal inversion $ r_1 = \dfrac{a^2}{r_2}, $ but by means of a Lens formula known from time of Gauss: $$ 1/r_1 + ...
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0answers
54 views

Derivative of $(\lambda I - A)^{-1}$ with respect to $\lambda$

Is need to work with $\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u)$. Is it true that: $$\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u) = -v^{T}\frac{d}{d\lambda}(\lambda I - A)^{-1}u$$ ...
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0answers
27 views

Finding the inverse of linear transformation using matrix

Assuming I have a linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation? I know I should ...
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1answer
661 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
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17 views

Find $v$ that maximises $v^TA\left(I_m-\sum_{i=1}^{p}B_i^Tvv^TB_i\right)^{-1}A^Tv$

I am trying to generate a rank 1 update rule for an optimization problem and I reach a point where I don't know how to continue. Let $v\in \mathbb{R}^n$ such that $\|v\|_2=1$ and also consider that ...
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Inverse of Covariance Matrix transforms sphere to ellipsoid?

sphere: f(X) = X^T X ellipsoid: f(AX) = X^T A^2 X Why is A^2 the inverse of the covariance matrix ? In other words, A is the covariance matrix to the power of -0.5.
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2answers
23 views

Congruent matrices - why do we require invertiblility?

If $A$, $B$ $\in K^{n \times n}$ are $n \times n$ matrices over a field $K$, then we say that $A$ and $B$ are congruent if there exists an invertible $P \in GL(n, K)$ such that $B = P^TAP$, where ...
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0answers
14 views

How can I convince myself of the Fourier scaling property via inverse FT?

I have this function $f(at)$, and I want to Fourier-tranform it. I proceed in the following way, for $\quad\alpha<0 \Longrightarrow a=-|a|$: \begin{align} \ \mathcal{F}_{t \rightarrow \xi}[f(at)]= ...
2
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1answer
34 views

Is there a way to update the inverse of a sum of two matrices following a rescaling of one of them?

Suppose I have two matrices $A$ and $B$ (let's assume that both $A$ and $B$ are invertible, as is their sum), and a scalar $g$. I am interested in the matrix $$M^{-1} = (A + gB)^{-1}$$ I am aware ...