# Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

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### Deriving the "Explicit" formula for inverse of Hilbert/Cauchy matrices

My exact question is, how to derive the formula for $H^{-1}$, in which $H_{ij}=\frac{1}{i+j-1}$. I am currently working my way through Hoffman&Kunze Linear Algebra. I noticed that a question on ...
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### update rule for the inverse after a rank-1 update plus scaled identity

Is there an update rule for $$\left(\tilde{X}^T\tilde{X}+\alpha\cdot I\right)^{-1}$$ with $\tilde{X}=[X\;\; a]$ as a function of $A\triangleq (X^TX)^{-1}$, $X$ and $a$? I know that when $\alpha=0$ we ...
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### Matrix inversion inequality

Suppose $A, B, C\in\mathbb{R}^{n\times n}$ are all symmetric positive definite matrices, and they satisfy the inequality $A \succeq B + C$. Assume also that all of the three matrices are bounded, i.e.,...
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### Complexity of pseudoinverse of a low-rank matrix

I have a large matrix $\mathbf{A}\in \mathbb{C}^{m\times n}$ with very low rank $r$, I find that the general complexity of finding its pseudoinverse is $\mathcal{O}(\max(m,n)^3)$, this is too high. ...
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### Inverting lower triangular matrix in time $n^2$

I have a lower $n\times n$ triangular matrix called $A$ and I want to get $A^{-1}$ solved in $O(n^2)$. How can I do it? I tried using a method called "forward substitution", but the ...
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I have a matrix with specific form of $A\otimes I + B\otimes J$ where $A$ and $B$ are general dense matrices, $n\times n$. $I$ is an $m\times m$ identity matrix. $J$ is a $m \times m$ dense matrix ...