Linear Algebra, Theory And Applications, 2012a

190 SPECTRAL THEORY
Proof: In the proof of Corollary 7.9.5, note that a ii is a simple root of A (0) since
otherwise the i th Gerschgorin disc would not be disjoint from the others. Also, K, the
connected component determined by a ii must be contained in D i because it is connected
and by Gerschgorin’s theorem above, K ∩ σ (A (t)) must be contained in the union of the
Gerschgorin discs. Since all the other eigenvalues of A (0) , the a jj , are outside D i , it follows
that K ∩ σ (A (0)) = a ii . Therefore, by Lemma 7.9.6, K ∩ σ (A (1)) = K ∩ σ (A) consists of
a single simple eigenvalue.
Example 7.9.8 Consider the matrix
⎛
⎝ 5 1 1 1 0
1
⎞
⎠
0 1 0
The Gerschgorin discs are D (5, 1) ,D(1, 2) , and D (0, 1) . Observe D (5, 1) is disjoint
from the other discs. Therefore, there should be an eigenvalue in D (5, 1) . The actual
eigenvalues are not easy to find. They are the roots of the characteristic equation, t 3 − 6t 2 +
3t +5=0. The numerical values of these are −. 669 66, 1. 423 1, and 5. 246 55, verifying the
predictions of Gerschgorin’s theorem.
7.10 Exercises
1. Explain why it is typically impossible to compute the upper triangular matrix whose
existence is guaranteed by Schur’s theorem.
2. Now recall the QR factorization of Theorem 5.7.5 on Page 133. The QR algorithm
is a technique which does compute the upper triangular matrix in Schur’s theorem.
ThereismuchmoretotheQR algorithm than will be presented here. In fact, what
I am about to show you is not the way it is done in practice. One first obtains what
is called a Hessenburg matrix for which the algorithm will work better. However,
the idea is as follows. Start with A an n × n matrix having real eigenvalues. Form
A = QR where Q is orthogonal and R is upper triangular. (Right triangular.) This
can be done using the technique of Theorem 5.7.5 using Householder matrices. Next
take A 1 ≡ RQ. Show that A = QA 1 Q T . In other words these two matrices, A, A 1 are
similar. Explain why they have the same eigenvalues. Continue by letting A 1 play the
role of A. Thus the algorithm is of the form A n = QR n and A n+1 = R n+1 Q. Explain
why A = Q n A n Q T n for some Q n orthogonal. Thus A n is a sequence of matrices each
similar to A. The remarkable thing is that often these matrices converge to an upper
triangular matrix T and A = QT Q T for some orthogonal matrix, the limit of the Q n
where the limit means the entries converge. Then the process computes the upper
triangular Schur form of the matrix A. Thus the eigenvalues of A appear on the
diagonal of T. You will see approximately what these are as the process continues.
3. Try the QR algorithm on (
−1 −2
6 6
which has eigenvalues 3 and 2. I suggest you use a computer algebra system to do the
computations.
)
4. Now try the QR algorithm on ( 0 −1
2 0
)