Let a be an M * n matrix, we prove that there exists an n-order matrix B ≠ 0, so that ab = 0 if and only if R (a)

Let a be an M * n matrix, we prove that there exists an n-order matrix B ≠ 0, so that ab = 0 if and only if R (a)

Proof: (= >)
Because AB = 0, the column vectors of B are solutions of AX = 0
Because B ≠ 0, ax = 0 has nonzero solution
So r (a)

On the operation properties of rational exponential power!
(AB) ^ R = a ^ RB ^ R (a > 0, b > 0, R belongs to q)
Why should a and B be greater than 0 and less than 0?

If a = 0 or B = 0, the exponent is meaningless when r = 0, so AB cannot be zero
When r = 1 / 2, (AB) ^ 1 / 2 = a ^ 1 / 2B ^ 1 / 2, the inner number of the root sign must be greater than 0, so AB cannot be less than 0

What are the operational properties of powers

If the power of a base is to the power of a power, then the power of a base is to the power of a power