Mr. Simonds MTH 261 - Supplemental problems regarding diagonalization
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1. Explain
geometrically
why the rotation matrix
(
)
(
)
() ()
cos sin
sin cos
R
θ
θ
θ
θ
−
⎡
⎤
=
⎢
⎥
⎣
⎦ cannot possibly have
any real number eigenvalues for 0
θ
π
<
<.
HINT: Write down what it would literally mean if
λ
were a real number eigenvalue with
corresponding eigenvector x. What would this imply geometrically about
R
x?
2. Find an orthogonal diagonalization of the matrix
.38 .18 .06 .04
.18 .59 .04 .12
.06 .04 .47 .12
.04 .12 .12 .41
A
−−−
⎡
⎤
⎢
⎥
−−
⎢
⎥
=
⎢
⎥
−− −
⎢
⎥
−−
⎣
⎦
. Use your
calculator to find both the eigenvalues
and
eigenvectors. Do
not
use your calculator to find
1
P−.
3. Joe Moma was taking a test and was instructed to find an orthogonal diagonalization of the
matrix 29 18
45 28
B⎡⎤
=⎢⎥
−−
⎣⎦
. Joe wrote “bogus problem” in the blank and went on to the next
question. What was Joe Moma thinking?
4. Find the
33× matrix A whose eigenvalues are 110
λ
=
, 23 5
λ
λ
=
=− with eigenspace bases
()
10
3
b
asis 1
1
V
⎧⎫
⎡⎤
⎪⎪
⎢⎥
=⎨⎬
⎢⎥
⎪⎪
⎢⎥
⎣⎦
⎩⎭
and
()
5
42
basis 1 , 5
22
V−
⎧
⎫
⎡
⎤⎡ ⎤
⎪
⎪
⎢
⎥⎢ ⎥
=−
⎨
⎬
⎢
⎥⎢ ⎥
⎪
⎪
⎢
⎥⎢ ⎥
−
⎣
⎦⎣ ⎦
⎩⎭
. Verify your result (and
not
just by
checking the key!).
5. Find 4 different matrices that when squared result in the matrix
A
where 21 50
15 34
A−−
⎡⎤
=⎢⎥
⎣⎦
.
6. Two brine tanks are connected by two pipes. One of the pipes flows from tank A to tank B and
the other flows in the opposite direction. Each pipe flows at the rate of 2 gal/min. Tank A
holds 120 gal and tank B holds 240 gal. Initially there are 24 lb of salt in tank A and tank B is
pure water. Determine how much salt is in each of the tanks in the long run. Assume that once
the pipes start to flow the mixtures in each tank are always perfectly mixed.
7. Use orthogonal diagonalization to help you find the rotation equation for the conic section with
equation 22
27 18 3 3 0xxyyxy−+++=. Also find the rotation angle to the nearest 10th of
a degree.
