the matrix
[ a -b ]^n
[ b a ]
[ a -b ]^n
[ b a ]
-
As it stands, there's not really a nice method... however, you can write
a = r * cos(t)
b = r * sin(t)
for r >= 0 and t real numbers.
Then
[ a -b ]^n
[ b a ]
can be written as
r^n * ...
[ cos(t) . . -sin(t) ] ^n
[ sin(t) . . .cos(t) ].
The matrix you're taking the n'th power of here can be recognized as a rotation matrix through an angle t, so raising it to the nth power would just be a rotation through an angle n*t. It follows that the result can be written as
[ r^n cos(n*t) . . . . -r^n sin(n*t) ]
[ r^n sin(n*t) . . . .. r^n cos(n*t) ].
Of course, you could use the trigonometric angle-sum formulas to write each component back in terms of a and b, but it would be both longer and more difficult to see the connection between the original matrix and its power.
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By the way, the above is a fairly general method. The matrices of the form
[a . . -b]
[b . .. a]
are algebraically what we call "complex numbers". Whenever you multiply or divide complex numbers, it's often easiest to write them in "polar form" first, which is what I did above with the r and the t ['t' for theta].
a = r * cos(t)
b = r * sin(t)
for r >= 0 and t real numbers.
Then
[ a -b ]^n
[ b a ]
can be written as
r^n * ...
[ cos(t) . . -sin(t) ] ^n
[ sin(t) . . .cos(t) ].
The matrix you're taking the n'th power of here can be recognized as a rotation matrix through an angle t, so raising it to the nth power would just be a rotation through an angle n*t. It follows that the result can be written as
[ r^n cos(n*t) . . . . -r^n sin(n*t) ]
[ r^n sin(n*t) . . . .. r^n cos(n*t) ].
Of course, you could use the trigonometric angle-sum formulas to write each component back in terms of a and b, but it would be both longer and more difficult to see the connection between the original matrix and its power.
-----------
By the way, the above is a fairly general method. The matrices of the form
[a . . -b]
[b . .. a]
are algebraically what we call "complex numbers". Whenever you multiply or divide complex numbers, it's often easiest to write them in "polar form" first, which is what I did above with the r and the t ['t' for theta].