# Tagged Questions

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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### Can $A^{T}(AA^{T})^{-1}A$ be simplified?

Let $A$ is an $m\times n$ ($m<n$) real matrix with full positive entries and $\text{Rank}(A)=m$. Thus $(AA^{T})^{-1}$ is an $m\times m$ symmetric $M$-matrix since $AA^{T}$ is nonnegtive and ...
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### Show that $Y$ is invertible

Let X be a $40\times40$ matrix such that $X^3 = 2I$. I want to show that $Y= X^2 -2X + 2I$ is invertible as well. I tried working with the equations to see if I can get Y as a product of matrices ...
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### Given a finite metric space, are the matrices of triangle inequality errors invertible?

I have been working on some problems regarding finite metric spaces and have already proven/positively answered the following statement/question if the underlying metric has additional properties. Now ...
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### Statement about $(I-A)^{-1}$ matrices

Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix. Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is such a diagonal ...
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### Prove a matrix expression leads to an invertible matrix?

I want to prove matrix $C$ is invertible: $$C=I-A^TB(B^TB)^{-1}B^TA(A^TA)^{-1},$$ where $I$ is an identity matrix of appropriate dimensions, and $(A^TA)^{-1}$ and $(B^TB)^{-1}$ imply both $A$ and $B$ ...
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### Which graphs do have invertible adjacency matrices?

I would like to know if there is any class of graphs for which the adjacency matrices are invertible. At this moment I am aware of only the class of graphs $n K_2$ which is the disjoint union of $n$ ...
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### Inverse of $\frac{1-e^{-x}}{x}$ on $(0,1)$

I am trying to invert (or to estimate the inverse of) $$y=\frac{1-e^{-x}}{x}$$ for $y\in(0,1)$. The function 'looks' monotonically decreasing between $x=0$ and $x=\infty$, but I have not been able to ...
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### Inverse of a Function exists iff Function is bijective

How to mathematically prove that inverse of a function, let's say, $f^{-1}$, exists, if and only if $f$ is bijective? I know how to prove it using diagrams but I'm looking for a rather mathematical ...
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### Finding the inverse of a map from $CP^1$ to $S^2$

Given the map: $$f:CP^1 \to S^2\ ,\ f[z:w] = \left(\frac{2\mbox{Re}(w\bar{z})}{|w|^2+|z|^2},\frac{2\mbox{Im}(w\bar{z})}{|w|^2+|z|^2}, \frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$ How would I go about ...
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### Super logarithmic inverse of tetration

What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$? Is it $slog^{x}_{2}$?
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### How to invert this function? (Inverse exponential function with arctan)

How to invert this function? $$y = e^{\arctan(x^5)}$$
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### The relation between an exponential function and a logarithmic function

I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're ...
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### An ill-conditioned matrix

If C is an ill-conditioned matrix and I want to get the inverse, one way is to take a pseudo-inverse of some sort. Instead, is the following, which uses the (normal) inverse, also a way to deal with ...
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### If matrix A is invertible, is it diagonalizable as well?

If a matrix A is invertible, then it is diagonalizable. Is it true or false?
$1 \frac{1}{2} ... \frac{1}{n}$ $\frac{1}{2} \frac{1}{3} ... \frac{1}{n+1}$ $.$ $.$ $.$ $\frac{1}{n} \frac{1}{n+1} ... \frac{1}{2n-1}$ Does the inverse of this matrix ...
### Solve equation $\tfrac 1x (e^x-1) = \alpha$
I have the equation $\tfrac 1x (e^x-1) = \alpha$ for an positive $\alpha \in \mathbb{R}^+$ which I want to solve for $x\in \mathbb R$ (most of all I am interested in the solution $x > 0$ for \$\...