A professor teaching linear algebra starts with a square matrix G whose QR decomposition is given by . He then defines matrices which has a QR decomposition of the form and a matrix . Given this, he asks the students to answer the following:

i) Do the matrices G and H have the same eigenvalues? Justify.

ii) Prove of disprove that G and M have the same eigenvalues.

__Solution__

What we know from the question:

(1)

(2)

(3)

(4)

i) Consider (1),

is orthonormalized form of G.

(5)

(6)

(7)

(8)

*(8) is a form of diagonalization and implies that H and G are similar matrices. I.e. H and G have the same eigenvalues.*

ii) Consider (3), ,

is orthonormalized form of G. Thus,

(9)

Apply (9) in (4) =>

(10)

(11)

*(11) => M and H are similar and have the same eigenvalues. Combine this with (8) and it means that G and M have the same eigenvalues.*