MATLAB: A matrix component raised to power has a complex result

complexmatrixpower

Hello,
I need to do a complex calculation with numerous constants that may change regarding initial conditions. To do so, I decided to separate the arithmetic form (Pi5) from these constants, and put them into a matrix (A5). But it returns a complex results
With tests, I identified the problem as shown below in the code. The complex result (B_matrix) comes out when I raise a matrix component to power -1.2895 (result from 'A_matrix').
The calculation using numerical values has the right answer. But the same calculation using matrix components (A5(3) and A_matrix) gives a complex result.
Can somebody explain me what is the difference between the two cases please ? How can I change my code to still use the matrix and have a non-complex result ?
hmax=862.65e-9;
hr = 814.23e-9;
A5=[1.61217; 1.13111; -1.74756; -1.49291; 2.535334; -0.075187; 1.135865];
Pi5= A5(1) * ( A5(2) + A5(3) ^ ( A5(4) .*((hr/hmax) . ^ (A5(5)))) + ...
     A5(6) .* (hr/hmax) .^ A5(7) );
% Test 1 : A5(4).*( (hr/hmax).^(A5(5))
A_num= (-1.49291*(hr/hmax)^2.535334)
A_matrix = (A5(4)*((hr/hmax).^(A5(5))))
%Test 2 : A5(3)^( A5(4).*( (hr/hmax).^(A5(5)) )
B_num= -1.74756^(-1.49291*(hr/hmax)^2.535334)
B_matrix =A5(3)^A_matrix
Command Window
A_num =
   -1.2895
A_matrix =
   -1.2895
B_num =
   -0.4868
B_matrix =
-0.2990 + 0.3842i

Best Answer

There's nothing surprising there. Raising a negative number to a fractional power often results in complex numbers. In fact, that's even the definition of 1i, it's -1 to the power of 1/2 (also known as the square root)
 (-1)^0.5  %a.k.a sqrt(-1), a.k.a 1i
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