Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 31
13.10 Matrix Functions
Distinct eigenvalue matrix functions
As shown above in the section on matrix eigensystems, we can express a matrix with
distinct eigenvalues in the form:
A � X � L � X �1
Hence we can write its exterior square as:
A 2 � �X � L � X �1 ���X � L � X �1 � � X � L ��X �1 � X��L � X �1 � X � L 2 � X �1
This result is easily generalized to give an expression for any power of a matrix.
A p � X � L p � X �1
Indeed, it is also valid for any linear combination of powers,
� a p �A p � � ap �X � L p � X �1 � X ��� ap �L p ��X �1
and hence any function f[A] defined by a linear combination of powers (series):
f�A� � � ap �A p
Now, since L is a diagonal matrix:
f�A� � X � f�L��X �1
L � DiagonalMatrix��Λ1 , Λ2 ,…,Λm ��
its pth power is just the diagonal matrix of the pth powers of the elements.
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L p � DiagonalMatrix��Λ1 p , Λ2 p ,…,Λm p ��
13.11
13.12