10.34 – Fall 2006
Homework #
4
 Solutions
Problem 1: Problem 3.A.3 in Beers’ textbook – Eigenvalue/vectors
Part A:
(see page 150151 in Beers)
We are asked is any of the matrices are known to have all real eigenvalues.
Matrices A
can be visually seen to have only real eigenvalues, this is because it is a real symmetric
matrix.
Matrix C can also be determined to have only real eigenvalues because:
Trace(C) – 4*det(C) = 24 > 0
This requires a small calculation, but not the actual calculation of the eigenvalues.
Part B:
(see page 156157 in Beers)
The bounding of the eigenvalues involves the use of Gershgorin’s Theorem, which states
that the eigenvalues of a matrix must lie within some range around the diagonal elements
of the matrix.
More specifically:
λ
i
−
a
kk
≤
∑
= Γ
k
a
k
j
elements j
≠
k
The summation is the sum of the modulus of all the offdiagonal elements for row
k
.
Matrix A:
Doing this analysis for matrix A, we find that the Gamma value for rows 1 – 4 are: 4, 5,
2, and 5, respectively. This results in the following bounds:
≤
4
or
4
λ
i
4
⎫
−
≤
≤
λ
i
⎪
λ
i
−
≤
2
5
or
3
λ
i
⎪
−
≤
≤
7
⎬
⇒
6
λ
i
7
−
≤
≤
λ
i
−
≤
3
3
or
0
λ
i
⎪
≤
≤
6
λ
i
+
≤
1
5
or
6
λ
i
⎭
⎪
−
≤
≤
4
Matrix C:
Similarly for matrix C, we find Gamma values of: 2 and 1.
3
2
or
1
λ
i
⎫
⎪
λ
i
−
≤
≤
≤
5
⎬
⇒
−
2
≤
≤
λ
i
5
1
1
or
2
λ
i
⎪
⎭
λ
i
+
≤
−
≤
≤
0
Note that in this case, the range between 0 and 1 is also excluded from the value
eigenvalue space, but the problem asked for an upper and lower bound.
Part C:
For a unitary matrix,
D
1
=
D
T
. So if we can calculate the inverse and show it equals the
transpose, then
D
is unitary.
In order to determine the inverse, we can make an
augmented matrix to solve
D
*
D
1
=
I
. Similar to the idea of Gaussian elimination, we
want to do row operations to turn
D
into
I
, and the matrix that was formerly
I
, will be the
inverse of
D
. To accomplish this, we only need to do two things: exchange rows 1 and 2;
then multiply row 2 by 1.
The results are shown below.
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to
Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu),
Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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⎡
0
−
1
0
⎢
1
0
0
⎢
⎢
0
0
1
⎣
1
0
0
⎤
⎡
1
0
0
⎥
Row operations
⎢
0
1
0
⎯⎯⎯⎯⎯→
0
1
0
⎥
⎢
0
0
1
⎥
⎦
⎢
⎣
0
0
1
0
1
0
⎤
−
1
0
0
⎥
⎥
0
0
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 Fall '04
 ClarkColton
 Linear Algebra, Matrices, Orthogonal matrix, Numerical Methods Applied

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