Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

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126 votes
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Should the formula for the inverse of a 2x2 matrix be obvious?

As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
Frank Thorne's user avatar
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4 votes
1 answer
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optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint. $D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter. $x$ and $v$ are two known p-dimensional vectors. The ...
Bing's user avatar
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26 votes
6 answers
13k views

Deriving inverse of Hilbert matrix

The Hilbert matrix is the square matrix given by $$H_{ij}=\frac{1}{i+j-1}$$ Wikipedia states that its inverse is given by $$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-...
L.Z. Wong's user avatar
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11 votes
2 answers
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Inverse of a small submatrix

Let $A$ be a large matrix (say, $1000 \times 1000$), and let $\mathcal I = \{2,3,5\}$ be a set of row/column indices. Let $(A^{-1})_{\cal I \times I}$ denote the submatrix of $A^{-1}$ that consists of ...
John Smith's user avatar
10 votes
3 answers
819 views

Find the inverse of a matrix that is very similar to the Hilbert matrix

The standard Hilbert matrix $H$ is given by $$H_{ij}=\frac{1}{i+j-1},$$ and it has an inverse given for example in this MO question. Now I have encountered a matrix $M$ of similar form, namely, $$...
Dings's user avatar
  • 153
8 votes
1 answer
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Efficiently compute the trace of a sparse matrix times the inverse of a sparse matrix?

How can I efficiently compute $\mathrm{trace}(A(B^{-1}))$ where $A$ and $B$ are both sparse symmetric PSD $n \times n$ matrices, both with $O(n)$ non-zero entries? If it helps, the pattern of non-...
Jeff's user avatar
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7 votes
3 answers
1k views

How to calculate inverse of sum of two Kronecker products with specific form efficiently?

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 ...
J. Doi's user avatar
  • 71
7 votes
1 answer
779 views

Trace of inverse of random positive-definite matrix in high dimension?

Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
Goulifet's user avatar
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5 votes
1 answer
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Find the inverse of a more general matrix that is similar to the Hilbert matrix

In the last MO question , the following matrix is given: $$M_{ij}=\left[\frac{1+(-1)^{i+j}}{i+j-1}\right]$$ and its inverse has been discussed. Now the problem is further extended to a more general ...
Dings's user avatar
  • 153
5 votes
1 answer
405 views

Inverse of a Cauchy-like matrix

Consider $n\times n$ symmetric Cauchy-like matrix $M$ with elements $(M_{ij})_{i,j=1}^{n}$ given by $$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} \frac{...
Abhishek Halder's user avatar