A determinantal proof of the Craig-Sakamoto theorem
Ingram Olkin · Linear Algebra and its Applications · 1997
The Craig-Sakamoto theorem states that if A and B are symmetric matrices, then (a) |I − αA − βB| = |I − αA||I − βB| for all α, β if and only if (b) AB = 0. There are a number of proofs of this result, the most common based on expansions of the logarithm of (a). The present proof is elementary in that it depends only on determinantal conditions.