Do elementary row operations change eigenvalues?
1 Answer
Yes. For a given matrix
For instance, take the following matrix:
#color(green)(hatA = [(2,2),(0,1)])#
The eigenvalues are determined by solving
#\mathbf(hatAvecv = lambdavecv),#
such that
#= |[(lambda,0),(0,lambda)] - [(2,2),(0,1)]|#
#= |(lambda - 2, -2),(0,lambda - 1)|#
#= (lambda-2)(lambda-1) - 0#
From this we acquire the characteristic equation:
#=> color(green)((lambda - 2)(lambda - 1) = 0),#
And we get the eigenvalues
# => color(blue)(lambda = 1, 2),#
whose eigenvectors are...
#[(lambda - 2, - 2),(0,lambda - 1)][(v_1),(v_2)] = [(0),(0)]#
#= [(-1,-2),(0,0)][(v_1),(v_2)] = [(0),(0)]#
#[(lambda - 2, - 2),(0,lambda - 1)][(v_1),(v_2)] = [(0),(0)]#
#= [(0,-2),(0,1)][(v_1),(v_2)] = [(0),(0)]#
Of course, had you row-reduced
#hatA = [(2,2),(0,1)]#
#stackrel(1/2R_1; -R_2+R_1" ")(->)[(1,0),(0,1)],# where the notation
#cR_i + R_j# implies that#c# times row#i# is added to row#j# and the result is stored into row#j# .
That would give you the characteristic equation
However, without row-reduction, we had gotten two distinct eigenvalues: