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aznkid310
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Homework Statement
Let a be all real numbers, nxn. Prove that
a) rank(A) less than or equal to rank(exp(A))
b) rank(exp(A)-I) less than or equal to Rank(A)
Homework Equations
I'm new to math proofs and so don't really know where to start. Could someone point me in the right direction? Would I need to prove that A is diagonalizable and somehow proceed from there?
The Attempt at a Solution
rank(A) + dim N(A) = n, N(A) = nullspasce of A
This means rank(A) less than or equal to n
exp(A) = I + A + (1/2)A^2 +...+ (1/(r-1)!)*A^(r-1) Taylor Series Expansion
Using Sylvester's Inequality: [rank(A) + rank(exp(A)) -n ] less than or equal to rank(Aexp(A))
Aexp(A) = A + A^2 + (1/2)A^3 + ... + (1/(r-1)!)*A^r
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