# Tag Info

### Practical example of why it is not good to invert a matrix

Here is a quick example which is very practical related to memory usage in PDEs. When one discretizes the Laplace operator, $\Delta u$, for example, in the Heat Equation $$u_t = \Delta u + f(t,u) .$$...
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### Practical example of why it is not good to invert a matrix

Normally there are some principal reasons to prefer solve a linear system respect to use the inverse. Briefly: problem with the conditional number (@GoHokies comment) problem in the sparse case (@...
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### How to directly compute the inverse of an ill-conditioned dense matrix

Though it is a relatively rare situation when you actually have to calculate an inverse of the matrix, not all techniques were created equally. I would use the term badly-conditioned instead of ill-...
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### Why is matrix inversion unstable when svd is stable?

The big issue is the condition number, which is defined as the ratio of the largest and smallest singular values. Suppose we expect: $$S = \begin{bmatrix} 10^{-15}\\ &1 \end{bmatrix}$$ If we ...
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### What algorithm(s) do numpy and scipy use to calculate matrix inverses?

Documentation to numpy.linalg.inv and scipy.linalg.inv does not mention the algorithm used. Judging from the source, ...
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### Do I really need to invert this matrix

Since $$A = B(I-B)^{-1} = (I-B)^{-1}(I-B)B(I-B)^{-1} = (I-B)^{-1}B(I-B)(I-B)^{-1} =(I-B)^{-1}B$$ So you want to solve $$(I-B)A=B$$ You seem to need only the first three columns of $A$. Solve the ...
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### Fast algorithm for computing cofactor matrix

So, a cofactor matrix is a transpose of an adjugate matrix. I know of the following paper: G. W. Stewart, "On the adjugate matrix," Lin. Alg. Appl., vol. 283, no. 1–3, pp. 151–164, Nov. ...
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### Approximately, at any given time, what proportion of the world's total HPC resources are dedicated towards inverting matrices?

The number is almost certainly unverifiable because every HPC system's overall user community is different and because there are so many systems around. I think that the number is vastly smaller than ...
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