## Wednesday, November 7, 2012

### cold core (aka cooling flow)

this week's question was suggested by R. Schmidt (a fellow cosmologist):
$X$-ray observations suggest that the centres of clusters of galaxies are sometimes colder than the rest of the cluster. what's the physical reason for these cold cores? Would cold cores show up in the thermal Sunyaev-Zel'dovich effect?

bonus question: can you show that $\ln\det(A) = \mathrm{tr}\ln(A)$, or equivalently $\exp\mathrm{tr}(A) = \mathrm{det}\exp(A)$ for any non-singular square matrix A?

bonus question 2: what do you think, do you have more or less friends than your friends have on average?

1. bonus question: the relation is easiest shown in the eigenframe of $A$, because both the trace and the determinant are invariants under orthogonal transformations. the trace is the sum of all eigenvalues, and the determinant the product of the eigenvalues. Clearly, the logarithm of the determinant would then be the sum of the logarithms of the eigenvalues.