Could someone please let me know how I would go about accomplishing the following tasks. Thank you.
Let A = [[4,0,0],[0,1,3],[0,3,1]
a) Find the Eigenvalues of A.
(lambda - 4)(lambda - 4)(lambda + 2)
So the eigenvalues are 4, 4, and -2.
b) For each eigenvalue, find a basis for the corresponding eigenspace.
c) Use these bases vectors to construct an orthogonal matrix P.
d) Verify that is a diagonal matrix.
I prefer to use the definition of "eigenvector" directly: If 4 is an eigenvector for A, then there exist a vector, v, such that Av= 4v. Specifically,
That gives the four equations 4x= 4x, y+ 3z= 4y, and 3y+ z= 4z. The first equation is satisfied by any x, of course and the last two are the same as 3y= 3z or z= y. That is, any eigenvector, corresponding to eigenvalue 4, is of the form , showing the two basis vectors for the eigenspace clearly.
With eigenvalue -2, those equations are simply 4x= -2x, y+ 3z= -2y, and 3y+ z= -2z. The first equation says x= 0 and I will leave the others to you.