Proving A Must Be of Rank 2: The 2x2 Matrix Dilemma

Thread starter LukasMont
Start date Sep 14, 2021
Tags

Matrix Proof rank

Click For Summary

The discussion centers on the challenge of proving that matrix A must be of rank 2. Participants clarify that it is unnecessary to demonstrate this rank, as it is provided as a premise. There is confusion regarding the relevance of the rank condition, given that the theorem holds for any matrix, not just those of rank 2. Ultimately, the proof presented fulfills the requirements of the problem, despite the initial concerns. The conversation highlights the nuances of matrix rank in the context of mathematical proofs.

Sep 14, 2021

LukasMont

Homework Statement: 4.1 Show that one may express any second rank matrix as the sum of a symmetric
and an antisymmetric matrix.

Relevant Equations: I was able to proof that any matrix could be constructed by adding a symmetric and antisymmetric matrix:

A= A/2 + A/2 + A'/2 - A'/2,
A= (A/2 + A'/2) + (A/2 - A'/2), where A' is the transposed matrix. Now,

A/2 + A'/2 is symmetric, since (A/2 +A'/2)' = A'/2 + A/2 (equal) and
A/2 - A'/2 is antisymmetric, since (A/2 - A'/2)' = - A'/2 + A/2= -(A/2 - A'/2).

My trouble is being to show A must be of rank 2. Any ideas?

Physics news on Phys.org

Sep 14, 2021

andrewkirk

Science Advisor

Homework Helper

Insights Author

Gold Member

You don't have to show that A has rank 2. You are given that as a premise.
I don't know why they give you that as a premise though, because the theorem is true for any matrix, not just any rank-2 matrix, as your proof shows.
Anyway, you have proven what they asked you to.