Zwiebach: Decompose Matrices w/Traceless Symmetric Part

[ehrenfest]

Homework Statement

On this page Zwiebach says that an arbitrary matrix can be decomposed into a traceless-symmetric part, an antisymmetric part, and the identity matrix times a constant. Isn't that only true when the original matrix has all off its diagonal elements equal, though?

EDIT: never mind; I figured it out-- the symmetric matrix matrix is traceless which does not mean that each diagonal element is 0, sorry

Homework Equations

The Attempt at a Solution

 

[Jhero]

Hello,

Thank you for your post. You are correct that the decomposition into a traceless-symmetric part, an antisymmetric part, and the identity matrix times a constant is only applicable when the original matrix has all of its diagonal elements equal. However, this is not the only case in which this decomposition can be applied.

The decomposition can also be used for any arbitrary matrix, even if the diagonal elements are not equal. In this case, the traceless-symmetric part will not necessarily have all of its diagonal elements equal to zero, but it will still be traceless (i.e. the sum of its diagonal elements will be zero). This means that the traceless-symmetric part can still be represented as a linear combination of the identity matrix and a constant.

I hope this clarifies any confusion you may have had. If you have any further questions, please don't hesitate to ask. Thank you for your interest in this topic.