4.14 Complex Eigenvalues
4.14.1 Matrices with Complex Eigenvalues
The version of the QR algorithm discussed in the previous sections won't approximate complex
eigenvalues. The reason is that complex eigenvalues of matrices with real entries occur in conju
Exam 1
Math 473/573
Winter 2004
Name: _ This is a closed book exam. You may use the formula sheet handed out
with the exam. Show all work and explain any reasoning which is not clear from the computations.
Turn in this exam with your answers.
1.
Let B be
Matrix Computations
Complex numbers
z = x yi
z = =
sign(z) =
if z = x + yi
Vectors
z* = (z1, zn) w* =
for z = w = (w1, , wn)
. w = inner product of z and w
z
for z = and w =
= z*w
= z1w1 + + znwn
z = =
for z =
z and w are orthogonal (perpendicular) if
Final Exam
Math 473/573
Winter 2004
Name: _ This is a closed book exam. You may use the formula sheet handed out
with the exam. Show all work and explain any reasoning which is not clear from the computations.
Turn in this exam with your answers.
1.
(15 p
4.15 The Eigenvalues of a 2 2 Matrix
In the version of the QR algorithm discussed in the next section we use as a shift the eigenvalue of the 2
2 submatrix in the lower right corner that is closest to the element in the lower right corner. In this sectio
4.16 The Gram Schmidt Process with Complex Vectors
When we use the QR algorithm with matrices of complex numbers we will be finding the QR factorization
of matrices with complex numbers. We can either use Gram Schmidt orthonormalization or Householder
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