Logarithm of a matrix: Difference between revisions

Revision as of 21:39, 2 January 2022

Revision as of 21:43, 2 January 2022

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A square matrix represents a [[linear operator]] on the [[Euclidean space]] '''R'''^''n'' where ''n'' is the dimension of the matrix. Since such a space is finite-dimensional, this operator is actually [[bounded operator|bounded]].

Using the tools of [[holomorphic functional calculus]], given a [[holomorphic function]] ''f''(''z'') defined on an [[open set]] in the [[complex plane]] and a bounded linear operator ''T'', one can calculate ''f''(''T'') as long as ''f''(''z'') is defined on the [[spectrum of an operator|spectrum]] of ''T''.

The function ''f''(''z'')=log ''z'' can be defined on any [[simply connected]] open set in the complex plane not containing the origin, and it is holomorphic on such a ___domain. This implies that one can define ln ''T'' as long as the spectrum of ''T'' does not contain the origin and there is a path going from the origin to infinity not crossing the spectrum of ''T'' (e.g., if the spectrum of ''T'' is a circle with the origin inside of it, it is impossible to define ln ''T'').