Inverting big matrices. REALLY BIG

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In summary, the conversation discusses methods for efficiently inverting large matrices, including finding the rank and null vectors, and potential ways to parallelize the process. The speaker expresses interest in finding a more efficient algorithm for matrix multiplication and acknowledges the potential challenge of maintaining numerical stability.
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Okefenokee
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Inverting big matrices. REALLY BIG!

How is it done?

Let's say I have a sparsely populated 1 gazillion by 1 gazillion square matrix in a formula like this A*x = b. What sort of efficient methods exist to do the following?: find the rank, invert it if it has full rank, find the null vectors if it does not have full rank.

Also, it would be great if the technique lends itself to parallel programing. It's for a pet project that I'm playing with. I can think of a couple ways to do this but I'm sure my solution would be sloppy compared to what is already out there. One of my ideas was to package the inner matrices into manageable blocks and do some Gaussian elimination in stages.
 
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It all depends on we can use further properties of the matrix. In general I would look for an efficient algorithm for matrix multiplication. I'm not sure where the current record holder for the matrix exponent lies, but I guess numerical stability is more of a problem than to implement an algorithm which only works for really big matrices, the more as the I/O operations outperform calculation times by far.
 

FAQ: Inverting big matrices. REALLY BIG

How do you define a "big" matrix in the context of matrix inversion?

In general, a "big" matrix refers to a matrix with a large number of rows and columns. The exact size that qualifies as "big" may vary depending on the specific problem or application, but it is typically considered to be hundreds or thousands of rows and columns.

What is the purpose of inverting a big matrix?

Inverting a matrix is a mathematical operation that involves finding the inverse matrix, which is a matrix that, when multiplied by the original matrix, results in the identity matrix. Inverting a big matrix is often necessary in various fields of science and engineering for tasks such as solving systems of equations, performing data analysis, and solving optimization problems.

Is it always possible to invert a big matrix?

No, it is not always possible to invert a big matrix. In order for a matrix to be invertible, it must be square (equal number of rows and columns) and have a non-zero determinant. If a big matrix does not meet these criteria, it cannot be inverted.

What are some common methods for inverting big matrices?

There are several methods for inverting big matrices, including Gaussian elimination, LU decomposition, and Cholesky decomposition. These methods involve various mathematical operations and algorithms to efficiently compute the inverse of a big matrix.

How can I efficiently compute the inverse of a really big matrix?

In order to efficiently compute the inverse of a really big matrix, it is important to use specialized software and algorithms that are designed for handling large matrices. Additionally, techniques such as parallel computing and matrix sparsification can also be used to improve the efficiency of computing the inverse of a really big matrix.

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