# Proof that norm of submatrix must be less than norm of matrix it's embedded in

• Simfish
In summary, the conversation discusses the problem of finding the maximum norm of a submatrix B of a matrix A, where A is a 4x3 matrix and B is a 1x1 matrix. The participants suggest using block matrices M_1 and M_2, and finding their norms, as well as embedding B in 4x4 block matrices. The conversation also mentions the use of the inequality ||ABC||\leq ||A||\,||B||\,||C|| to solve the problem.
Simfish
Gold Member

## Homework Statement

http://dl.dropbox.com/u/4027565/2010-10-10_194728.png

## The Attempt at a Solution

||B|| = ||M_1 * A * M_2 ||

So from an equality following from the norm, we can get...

||B|| <= ||M_1||*||A||*||M_2||.

Now, we know that B is a submatrix of A. So if A is 4x3, then M_1 must be 1x4 and M_2 must be 3X1 (I know that block matrices are more complicated than that, but this might work). What this also means is that the combined product of M_1 and M_2 must be <= 1. But beyond that, I'm stuck. Is there another step I should take?

Thanks!

Last edited by a moderator:
Okay so if B is ultimately 1x1, then M_1 and M_2 must both be matrices with 0s everywhere except for one row (or column). So the maximum norm (under any situation) would be 1.

You cane always embed B in 4x4 block matrices by adding 0 and I as appropriate blocks. Block matrices M_1 and M_2 will probably be orthogonal projections of norm ||M||=1. In any case the key is to find explicitly M_1 and M_2 and their norms. Then use $$||ABC||\leq ||A||\,||B||\,||C||.$$

Last edited:

## Related to Proof that norm of submatrix must be less than norm of matrix it's embedded in

1. What is the definition of a submatrix?

A submatrix is a smaller matrix that is created by selecting a subset of rows and columns from a larger matrix. This subset must maintain the same order and arrangement as the original matrix.

2. Why is the norm of a submatrix important?

The norm of a submatrix is important because it measures the size or magnitude of the submatrix. It can be used to determine the similarity or difference between two matrices, and it is also a key factor in many mathematical calculations.

3. How is the norm of a submatrix related to the norm of the matrix it is embedded in?

The norm of a submatrix is always less than or equal to the norm of the matrix it is embedded in. This is because the submatrix contains fewer elements and thus has a smaller total value than the original matrix.

4. Can the norm of a submatrix ever be greater than the norm of the matrix it is embedded in?

No, the norm of a submatrix can never be greater than the norm of the matrix it is embedded in. This is a mathematical property that is always true, regardless of the specific values or elements within the matrices.

5. What implications does this proof have in the field of mathematics and science?

This proof has important implications in many areas of mathematics and science, particularly in linear algebra and matrix computations. It helps to establish the relationship between submatrices and their parent matrices, and can be used to make more accurate calculations and predictions in various fields of study.

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