EGR 6013 Homework Set 2 (50-55)
— David Wagner 2009/09/20 01:09
Problem 50.
(a) Determine the characteristic numbers (λ1, λ2) and corresponding unit characteristic vectors ({e1}, {e2}) of the matrix
![Lambda = delim{[}{ matrix{2}{2}{ 5 2 2 2 } }{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_c128452e02a7de2ac5d003722967345b.png "Lambda = delim{[}{ matrix{2}{2}{ 5 2 2 2 } }{]}")
.
(b) Verify that {e1} and {e1} are orthogonal.
© if {v} = {1, 1}, determine α1 and α1 so that
.
(d) Use the results of (a), together with equation (105), to obtain the solution of the following set of equations:
,
.
Consider the exceptional case separately.
(105) 
Problem 51.
(a) Suppose that the n characteristic vectors of the real symmetric matrix [A] are not normalized (reduced to unit length). If they are denoted by {u1}, {u2}, … ,{un}, show that (105) must be replaced by the equation
.
(b) Verify this result in the case of Problem 50(d).
Section 1.13.
√Problem 54.
Construct a set of three mutually orthogonal unit vectors which are linear combinations of the vectors {1, 0, 2, 2}, {1, 1, 0, 1}, {1, 1, 0, 0}.
.
![v_2 =
delim{lbrace}{matrix{4}{1}{ 1 1 0 1
} }{rbrace}
-
delim{]}{matrix{1}{4}{ {sqrt{2}/2} {sqrt{2}/2} {0} {0}
} }{[}
delim{lbrace}{matrix{4}{1}{ 1 1 0 1
} }{rbrace}
delim{lbrace}{matrix{4}{1}{ {sqrt{2}/2} {sqrt{2}/2} {0} {0}
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_901.5_5f682a3f7894f973d6749bf32151d189.png "v_2 =
delim{lbrace}{matrix{4}{1}{ 1 1 0 1
} }{rbrace}
-
delim{]}{matrix{1}{4}{ {sqrt{2}/2} {sqrt{2}/2} {0} {0}
} }{[}
delim{lbrace}{matrix{4}{1}{ 1 1 0 1
} }{rbrace}
delim{lbrace}{matrix{4}{1}{ {sqrt{2}/2} {sqrt{2}/2} {0} {0}
} }{rbrace}")
=
![v_1 =
delim{lbrace}{matrix{4}{1}{ 1 0 2 2
} }{rbrace}
-
delim{]}{matrix{1}{4}{ {sqrt{2}/2} {sqrt{2}/2} {0} {0}
} }{[}
delim{lbrace}{matrix{4}{1}{ 1 0 2 2
} }{rbrace}
delim{lbrace}{matrix{4}{1}{ {sqrt{2}/2} {sqrt{2}/2} {0} {0}
} }{rbrace}
-
delim{]}{matrix{1}{4}{ 0 0 0 1
} }{[}
delim{lbrace}{matrix{4}{1}{ 1 0 2 2
} }{rbrace}
delim{lbrace}{matrix{4}{1}{ 0 0 0 1
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_901.5_f98bfd71b3d0e2915eaaab9ecaf8b08f.png "v_1 =
delim{lbrace}{matrix{4}{1}{ 1 0 2 2
} }{rbrace}
-
delim{]}{matrix{1}{4}{ {sqrt{2}/2} {sqrt{2}/2} {0} {0}
} }{[}
delim{lbrace}{matrix{4}{1}{ 1 0 2 2
} }{rbrace}
delim{lbrace}{matrix{4}{1}{ {sqrt{2}/2} {sqrt{2}/2} {0} {0}
} }{rbrace}
-
delim{]}{matrix{1}{4}{ 0 0 0 1
} }{[}
delim{lbrace}{matrix{4}{1}{ 1 0 2 2
} }{rbrace}
delim{lbrace}{matrix{4}{1}{ 0 0 0 1
} }{rbrace}")
=
=
=
=
The three unit vectors e are
,
,
.
√Problem 55.
Prove that the vector {v} = {2, 1, 2, 0} is in the space generated by the three vectors defined in Problem 54 and express {v} as a linear combination of the selected vectors {e1}, {e2}, and {e3}.
This is true if {v} can be made from a linear combination of {ei}: [e]{c}={v}.
![delim{[}{matrix{4}{3}{
{sqrt{2}/6} 0 {sqrt{2}/2}
{-sqrt{2}/6} 0 {sqrt{2}/2}
{{2sqrt{2}}/3} 0 0
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ c_1 c_2 c_3
} }{rbrace}
=
delim{lbrace}{matrix{4}{1}{ 2 1 2 0
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_896.5_56fd25be864f115b91d47642cfd1b293.png "delim{[}{matrix{4}{3}{
{sqrt{2}/6} 0 {sqrt{2}/2}
{-sqrt{2}/6} 0 {sqrt{2}/2}
{{2sqrt{2}}/3} 0 0
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ c_1 c_2 c_3
} }{rbrace}
=
delim{lbrace}{matrix{4}{1}{ 2 1 2 0
} }{rbrace}")
![delim{[}{matrix{4}{3}{
{1} 0 {3}
{-sqrt{2}/6} 0 {sqrt{2}/2}
{{2sqrt{2}}/3} 0 0
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ c_1 c_2 c_3
} }{rbrace}
=
delim{lbrace}{matrix{4}{1}{ {12/sqrt{2} } 1 2 0
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_911.5_f25c4079874e18036911dbce2b52668c.png "delim{[}{matrix{4}{3}{
{1} 0 {3}
{-sqrt{2}/6} 0 {sqrt{2}/2}
{{2sqrt{2}}/3} 0 0
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ c_1 c_2 c_3
} }{rbrace}
=
delim{lbrace}{matrix{4}{1}{ {12/sqrt{2} } 1 2 0
} }{rbrace}")
![delim{[}{matrix{4}{3}{
{1} 0 {3}
{-sqrt{2}/6} 0 {{sqrt{2}}/2}
{{2sqrt{2}}/3} 0 0
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ c_1 c_2 c_3
} }{rbrace}
=
delim{lbrace}{matrix{4}{1}{ {6sqrt{2}} 1 2 0
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_911.5_9845564ad130f1be993a19e35f0661ed.png "delim{[}{matrix{4}{3}{
{1} 0 {3}
{-sqrt{2}/6} 0 {{sqrt{2}}/2}
{{2sqrt{2}}/3} 0 0
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ c_1 c_2 c_3
} }{rbrace}
=
delim{lbrace}{matrix{4}{1}{ {6sqrt{2}} 1 2 0
} }{rbrace}")
![delim{[}{matrix{4}{3}{
1 0 {3}
0 0 {sqrt{2}}
0 0 {-2sqrt{2}}
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ c_1 c_2 c_3
} }{rbrace}
=
delim{lbrace}{matrix{4}{1}{ {6sqrt{2}} 3 {-6} 0
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_926.5_b3712bf1d258063df971bfd026019089.png "delim{[}{matrix{4}{3}{
1 0 {3}
0 0 {sqrt{2}}
0 0 {-2sqrt{2}}
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ c_1 c_2 c_3
} }{rbrace}
=
delim{lbrace}{matrix{4}{1}{ {6sqrt{2}} 3 {-6} 0
} }{rbrace}")
⇒ 
⇒ 
⇒ 
![{v}
=delim{lbrace}{matrix{4}{1}{ 2 1 2 0
} }{rbrace}
=
delim{[}{matrix{4}{3}{
{sqrt{2}/6} 0 {sqrt{2}/2}
{-sqrt{2}/6} 0 {sqrt{2}/2}
{{2sqrt{2}}/3} 0 0
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ {{3sqrt{2}}/2} {0} {{{3sqrt{2}}/2}}
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_896.5_6492f5c3bfef809fbbc1602c95aee259.png "{v}
=delim{lbrace}{matrix{4}{1}{ 2 1 2 0
} }{rbrace}
=
delim{[}{matrix{4}{3}{
{sqrt{2}/6} 0 {sqrt{2}/2}
{-sqrt{2}/6} 0 {sqrt{2}/2}
{{2sqrt{2}}/3} 0 0
0 1 0
} }{]}
delim{lbrace}{matrix{3}{1}{ {{3sqrt{2}}/2} {0} {{{3sqrt{2}}/2}}
} }{rbrace}")
