# Traceless matrices

• 01-01-2014, 02:44 PM
Ayesha
Traceless matrices
Is it necessary for a traceless matrix to be even order?
Am working on Dirac equation and I wonder why gamma matrices should be even order and 4 by 4?
• 01-01-2014, 08:32 PM
tk421
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Is it necessary for a traceless matrix to be even order?

No. Think of it this way: The trace is the sum of the eigenvalues. There's no necessity for even-ness in order to have a zero eigenvalue sum. As a simple example, consider a third-order dynamical system with a symmetrical pair of eigenmodes (real, with values that are algebraic inverses), and a third eigenmode at zero.
• 01-03-2014, 03:41 AM
KJW
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Think of it this way: The trace is the sum of the eigenvalues.

For finite-dimensional matrices, isn't it simpler to regard the trace as the sum of its diagonal components?
• 01-03-2014, 06:18 AM
tk421
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For finite-dimensional matrices, isn't it simpler to regard the trace as the sum of its diagonal components?

Simpler, perhaps, but to my tastes, less informative than it could be. The trace is certainly the sum of the main diagonal components, so it's easy enough to see how one could come up with an infinite variety of combinations that sum to zero. I just prefer physically-based examples when possible, hence the connection to the eigenvalue sum. You say tomato, I say banana. :)