The inverse of a banded matrix

In summary, the conversation discusses the use of a banded matrix to save complexity in inverting a 512-by-512 matrix with most elements outside the diagonals between -5 and +5 being small. The suggestion is to instead decompose the banded matrix to preserve efficiency in solving a set of equations.
  • #1
EngWiPy
1,368
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Hello all,

I have say 512-by-512 matrix, but based on the structure of this matrix most elements not on the diagonals between -5 to +5 (- stand for diagonal below the main diagonal, and + for diagonal above the main diagonal) are small relative to the elements of the mentioned diagonals. So, I create a 512-by-512 banded matrix, where I null all other elements not on the mentioned diagonals.

Now the question is: will there be a huge complexity saving if I want the inverse of the matrix by inverting its banded version instead of the original matrix?

Thanks
 
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  • #2
In computer calculations, "inverting a matrix" is almost always the wrong thing to do, even if you have a nice looking math equation with an inverse matrix in it.

In this case there will be a huge "complexity" increase, because the inverse matrix will be fully populated, not banded.

What you really want to do is probably solve a set of equations or something similar. If you decompose your banded matrix as A = LDU or something similar, where L and U are lower and upper triangular and have the same bandwidth as A, you preserve the efficiency by not needing to process all the zero terms in L and U.
 
  • #3
AlephZero said:
In computer calculations, "inverting a matrix" is almost always the wrong thing to do, even if you have a nice looking math equation with an inverse matrix in it.

In this case there will be a huge "complexity" increase, because the inverse matrix will be fully populated, not banded.

What you really want to do is probably solve a set of equations or something similar. If you decompose your banded matrix as A = LDU or something similar, where L and U are lower and upper triangular and have the same bandwidth as A, you preserve the efficiency by not needing to process all the zero terms in L and U.

Could you please tell me more about this process?
 

1. What is a banded matrix?

A banded matrix is a type of square matrix in which the majority of the elements are zero, except for those within a certain distance from the main diagonal. This distance is called the bandwidth, and a matrix with a bandwidth of 1 is known as a tridiagonal matrix.

2. What is the inverse of a banded matrix?

The inverse of a banded matrix is another matrix that, when multiplied by the original banded matrix, results in the identity matrix. It is essentially the "opposite" of the original matrix, and can be thought of as undoing the operations performed by the original matrix.

3. How is the inverse of a banded matrix calculated?

The inverse of a banded matrix can be calculated using various methods, such as Gaussian elimination or LU decomposition. These methods take advantage of the special structure of banded matrices to efficiently compute the inverse.

4. What are the applications of inverse banded matrices?

Inverse banded matrices have various applications in fields such as engineering, physics, and computer science. They are commonly used in solving systems of linear equations, as well as in numerical methods for solving differential equations.

5. Can all banded matrices be inverted?

No, not all banded matrices can be inverted. The ability to calculate the inverse of a banded matrix depends on its specific structure and properties. In some cases, the inverse may not exist or may be difficult to compute due to factors such as ill-conditioning or numerical instability.

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