Show that if v is an eigenvector for a matrix A associated t

Show that if v is an eigenvector for a matrix A associated to an eigenvalue lambda, then w = Av is also an eigenvector for a matrix A associated to lambda.

Solution

Let v be an eigenvector of the matrix A associated with the eigenvalue . Then (A – I)v = 0 or, Av –Iv = 0 or Av = Iv = v . Let w = Av . Then w = v. Also (A –I)w = ( A- I)v= [(A – I)v] = *0 = 0. Therefore w = v is an eigenvector of A associated with the eigenvalue .