# Questions tagged [inverse]

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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### Interesting Simple Integral Over the Unit Interval of the Inverse Regularized Incomplete Gamma Function. Non-Integral Form Needed. Closed Form?

I have recently used, $\Bbb {here}$, with the Regularized Incomplete Gamma Function. This then made me wonder about its inverse. This function can easily be integrated with respect to its second ...
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### Given two bases vectors for $\mathbb C_3$ find change of bases matrix from $e_1,e_2,e_3$ to $a_1,a_2,a_3$.

Given two bases vectors for $\mathbb C_3$ find change of bases matrix from $e_1,e_2,e_3$ to $a_1,a_2,a_3$. What we did in class was to stack $e_1, e_2, e_3$ into columns of $A$ and $a_1,a_2,a_3$ into ...
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### $A$ orthogonal matrix with the eigenvalue not equal to -1. Proving A is expressible as $(I+S)(I-S)^{-1}$ where $S$ is skew-symmetric matrix. [closed]

If $A$ be an orthogonal matrix with the eigenvalue not equal to $1$. Prove that $A$ is expressible as $(I+S)(I-S)^{-1}$ where $S$ is skew-symmetric matrix.
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### Shifting a positive semi-definite matrix

Let $n\in \mathbb{N}$, and $A\in \mathbb{R}^{n\times n}$ be a semi-positive definite matrix. What can we say about the matrix $$A_h:= A+(h-1)I,$$ where $I$ is the identity matrix, and $h>0$ (...
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### Inverse matrix calculation/transformation

When does the following hold? $$\frac{v^{\top} A v}{v^{\top} v} = \frac{v^{\top} v}{v^{\top} A^{-1} v}$$
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### How to extract B from ABA^T where A is not square

So I have this value in the form of ABA^T where A is 3 by 2 and B is 2 by 2. I want to retrieve B, but since A is not square, it does not have an inverse. Is there a way to do it?
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### Getting an equation involving logarithm into explicit form

I am at the final part of a problem where I have derived $t+c=\frac{1}{\sqrt2}\log\left(\frac{x}{2+\sqrt{4-2x^2}}\right)$ where $c$ is a constant, and now I need to express it in explicit form $x(t)$. ...
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### Inverse matrix of a 2n x 2n matrix given below.

I have trouble on finding the inverse matrix of $V$, given below. I tried finding first the inverse matrix to the case where $n=2,3$. But, I cant find a pattern that will lead me to the general one. ...
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### If $A$ is non-singular, prove $|A| = \frac{1}{|A^{-1}|}$? [duplicate]

I am unsure on how to prove the following problem: If a $2 \times 2$ matrix $A$ is non-singular, prove $|A| = \frac{1}{|A^{-1}|}$ I know that $|A|\cdot|A^{-1}| = I$ but i’m not sure where to go from ...
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### Approximate matrix inverse by Fourier approach

Given a hermititan matrix $A$ with the possibility to generate $e^{-iAt}$ for $t\geq 0$ how would I proceed to approxiamte: $A^{-1}\approx\sum_j\alpha_je^{-iAt_j}$ $\quad$ with $\alpha_j\in\mathbb{C}$...
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### When does one invertibility condition suffices?

It is often the case that, to prove $f$ being the inverse morphism of $g$, one has only to show $fg = id$ and the other direction ($gf = id$) is guaranteed to be true -- e. g. when considering vector ...
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### Is the inverse of a restricted compact operator unbounded?

Suppose we have two separable Hilbert spaces $\mathbb{H}_{1},\mathbb{H}_{2}$ and the compact operator $\mathscr{T}:\mathbb{H}_{1}\to\mathbb{H}_{2}$. We know that since $\mathscr{T}$ is compact, its ...
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### Inverting a Block-Toeplitz matrix with the Sherman-Morrison formula

Suppose we are given the following Block-Toeplitz matrix: \begin{eqnarray} T=\left(\begin{matrix} A & 0 & ... & 0\\ B & A & ... & \vdots\\ \vdots & \ddots & \ddots &...