MATLAB: Number^matrix plz help me on this, I cant understand this, plz help me

matrix and powerquestion_of_the_week

number^matrix example a=[1 2;3 4] is a 2×2 matrix 3^a= [87.8864,127.1198;190.6797,278.5661] how come these numbers? [87.8864,127.1198;190.6797,278.5661]
plz help me on this algorithm of calculation these numbers…

Best Answer

If the exponent was a scalar, say a=2, then :
a = 2;
3^2 = 9 = exp(2 * log(3))
If you define z = 2*log(3), then
3^2 = 9 = exp(z) = z^0 + z^1 + z^2/2 + z^3/6 + z^4/24 + z^5/120 + ...
The matrix exponential can be defined in the same way
a = [1 2; 3 4]
3^a = exp([1 2; 3 4] * log(3))
If you define z = a*log(3), then
3^a = z^0 + z^1 + z^2/2 + z^3/6 + z^4/24 + z^5/120 + ...
where a and z are now 2x2 matrices.
This infinite sum will converge to the answer. The actual method MATLAB uses is somewhat different (it doesn't use the Taylor series, it uses a Pade approximation) but the general result is the same.
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