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).

See also

Jordan Canonical Form, Matrix Power, Matrix Root, P-Matrix

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References

Higham, N. J. Functions of Matrices: Theory and Computation. Philadelphia, PA: SIAM, 2008. https://doi.org/10.1137/1.9780898717778.Xie, W. "Complete Bernstein Functions and Real Powers of -Matrices." 25 Oct 2026. https://arxiv.org/abs/2611.00096v1.

Cite this as:

Weisstein, Eric W. "Principal Matrix Power." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PrincipalMatrixPower.html

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