Unitary Block Diagonalization and the Characteristic Polynomial of a Pencil Generated by Hermitian Matrices

We consider the problem of simultaneously putting a set of square matrices into the same block upper triangular form with a similarity transformation, and obtain a result linking the size of the largest block to polynomial identities. This generalizes McCoy's theorem, which gives necessary and sufficient conditions for a set of matrices over an algebraically closed field to be simultaneously similar to upper triangular matrices. A theorem of Specht states that when the algebra a generated by a set of complex matrices satisfies the condition a = a*, where * denotes conjugate transpose, the algebra can be simultaneously, unitarily block diagonalized if and only if it can be simultaneously, block upper triangularized. Applying this to a single complex matrix A, we see that A can be unitarily block diagonalized if and only if A and A* can be simultaneously block triangularized. Let A = H + iK, where H and K are Hermitian. Then A can be unitarily block diagonalized if and only if H and K can be simultaneously block upper triangularized. Hence, to study the problem of unitarily, block diagonalizing the matrix A, we consider the pair of Hermitian matrices H and K. We study the pencil xH + yK and the characteristic polynomial of this pencil, f(x,y,z) = det(zI - xH - yK). Motzkin and Taussky proved that H and K commute if and only if f(x,y,z) factors into linear factors. If H and K can be simultaneously block diagonalized, then f(x,y,z) splits into factors corresponding to the blocks, but examples show that the converse is not true. However, we prove that if f(x,y,a) has a repeated linear factor of high enough multiplicity, then H and K have a common eigenvector. We also show that if f(x,y,z) is a power of e. quadratic polynomial, then H and K are simultaneously, unitarily similar to block diagonal matrices, each of which is a sum of identical 2 X 2 blocks. This is e. sharper version of a result of Kippenhahn. Motzkin and Taussky studied the algebraic curve of degree n whose equation is f(x,y,z) = O. Murnaghan and Kippenhahn independently showed that the curve whose equation in line coordinates is f(x,y,z) = O determines the numerical range of the matrix A. These geometric interpretations yield some information about A = H + iK when f(x,y,z) has only a linear factor and a quadratic factor. Kippenhahn conjectured that if f(x,y,z) has a repeated factor, then A is unitarily similar to a matrix which is block diagonal. We verify this conjecture for n ≤ 5.