But there are simply far too many areas of life involving putative "orthonormally diagonalizable matrices" for any one individual to be able to rationally investigate. At some point you have to take someone's word for it; so rather than taking one expert's word, you're likely better off trusting a community of experts. A current example might be with global warming - most scientists seem to feel it's a major issue.

Unfortunately, though, radical changes in thinking come usually come from the margin, e.g., Galileo. The hard part, it seems to me, is to distinguish between mere status quo convention and genuine expert agreement.