Let be an real matrix such that

(where I denotes the n × n-identity matrix). Show that

(i) A has two eigenvalues .

(ii) Every element can be expressed uniquely as ,

where and

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- March 6th 2009, 09:57 PMChandru1Eigenvalues
Let be an real matrix such that

(where I denotes the n × n-identity matrix). Show that

(i) A has two eigenvalues .

(ii) Every element can be expressed uniquely as ,

where and - March 7th 2009, 07:43 AMHardwarista
Take two endomorphisms of :

You have

according to the problem.

You see that

and, after a little manipulation and comaprison of dimensions,

dim Ker (u) + dim Ker (v) = n

Which is sufficient to say that

, cause it would mean v = 0

, cause it would mean u = 0

So there are effectively 2 eigenvalues for A ( 1 and -1),

and the associated eigenspaces are in direct sum (Ker u and Ker v),

which guarantees the existance and unicity if the decomposition.