Principal Matrix Power

The principal matrix power of a square complex matrix with no eigenvalue on is the matrix obtained by applying the branch , with , to the Jordan canonical form of (Higham 2008). Here is real, is the natural logarithm, and is the complex argument.

For a Jordan block of size , where , the definition is

where and for . For , it gives

The result is independent of the chosen Jordan canonical form and agrees with ordinary matrix powers for integer . This construction does not mean entrywise exponentiation.

A real P-matrix has no eigenvalue on , so all its principal real powers are defined. Such powers need not remain P-matrices in orders at least 4 (Xie 2026).