Articles of inverse

show invertibility of linear transformations

This question already has an answer here: Showing a Linear Transformation is invertible 2 answers

Finding only first row in a matrix inverse

Let’s say I have a somewhat large matrix $M$ and I need to find its inverse $M^{-1}$, but I only care about the first row in that inverse, what’s the best algorithm to use to calculate just this row? My matrix $M$ has the following properties: All its entries describe probabilities, i.e. take on values […]

Continuity of inverse function (via sequences)

If $f:[a,b] \rightarrow [c,d]$ is a continuous bijective function, then prove $f^{-1} $ (its inverse function) is continuous on $[c,d]$. I know this can be proven by using monotony of $f$, can anyone help me finish my approach? $\textbf{My approach: }$ Let $s_n \rightarrow s $ on $[c,d]$. And let $ t_n = f^{-1}(s_n),$ and […]

Inverse of matrix of ones + nI

Having a vector $\mathbf{1} \in \mathbb{R}^{n}$ containing only ones, following equality should be true according to a paper I am currently reading: \begin{equation} \left( nI+\mathbf{1}\mathbf{1}^T \right)^{-1}= \frac{1}{n}\left( I – \frac{1}{2n} \mathbf{1}\mathbf{1}^T \right) \end{equation} EDIT: what is the general rule for constructing an inverse of a matrix with $n$ on diagonal and $1$ elsewhere and how […]

Proof of the Sherman-Morrison Formula

I was reading a few proofs for the Sherman-Morrison Formula, which states that if $A$ is invertible and $M = A + \mathbf{u}\mathbf{v}^T$, then $M^{-1}$ is given by: $$A^{-1} – A^{-1}\mathbf{u} \mathbf{v}^T A^{-1}/(1+\mathbf{v}^TA^{-1}\mathbf{u}).$$ There is a proof (verification) of this on Wikipedia as well as here but both of them do not justify why $(1+\mathbf{v}^TA^{-1}\mathbf{u})$ […]

Multiplicative Inverse of a Power Series

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = -\frac{1}{p_0}\sum_{0 \leq i < n}p_{n-i}q_{i}$$ What’s the closed form of this recurrence? Writing out a bunch of terms hasn’t yet revealed to me […]

What is the inverse z transform of 1/(z-1)^2?

I’d like to know how to calculate the inverse z transform of $\frac{1}{(z-1)^2}$ and the general case $\frac{1}{(z-a)^2}$

Optimization that involves inverse operation.

$\newcommand{\diag}{\operatorname{diag}}$ I have the following optimization problem: \begin{align} \mathop{\arg\min}_\beta & \frac{1}{2} a’ [ M + \diag( \beta ) \otimes I_d ]^{-1} a + 1^T\beta, \quad \text{s.t. } \beta \geq 0 \end{align} where \begin{align} \beta &\in\mathbb{R^m } \, \text{is a column vector}\\ A &\in\mathbb{R^{m \times k}}, \,\, B\in\mathbb{R^{d\times k}}, \,\,I_d \text{ is an Identity matrix of […]

Inverse of Ulam's spiral

I have a program and I need a function that takes a coordinate as input and returns an integer corresponding to the position in Ulam’s spiral. The simple (but slow) way to do this would be to generate the spiral until you finally find a coordinate that matches the input and then return how many […]

Inverse of a matrix with $a+1$ on the diagonal and $a$ in other places

Let $a>0$. Let $A$ be the $n\times n$ matrix with $a+1$ on the diagonal and $a$ in all other entries. How can one compute $A^{-1}$ as a function of $n$?