Table of Contents
— David Wagner 2009/09/15 19:52
EGR 6013 Homework Set 1 (6-11)
Problem 6.
[A][A] = 0 = ![[a_ik][a_kj]](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_980.5_0ca98e5ecfb9678d5b828aa4cd0adaff.png "[a_ik][a_kj]")
= 
.
=![delim{[}{matrix{2}{2}{a_11 a_12 a_21 a2_2 }}{]}
delim{[}{matrix{2}{2}{a_11 a_12 a_21 a_22 }}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_2f5a762eb9bbb63eebaf852a10162940.png "delim{[}{matrix{2}{2}{a_11 a_12 a_21 a2_2 }}{]}
delim{[}{matrix{2}{2}{a_11 a_12 a_21 a_22 }}{]}")
=
![delim{[}{matrix{2}{2}{
{a_11^2 + a_12 a_21} {a_11 a_12 + a_12 a_22}
{a_21 a_11 + a_22 a_21} {a_21 a_12 +a_22^2 }
}}{]}
=
delim{[}{matrix{2}{2}{0 0 0 0 }}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_956.5_95fa4595e5254c5783efe341cf3a23dd.png "delim{[}{matrix{2}{2}{
{a_11^2 + a_12 a_21} {a_11 a_12 + a_12 a_22}
{a_21 a_11 + a_22 a_21} {a_21 a_12 +a_22^2 }
}}{]}
=
delim{[}{matrix{2}{2}{0 0 0 0 }}{]}")
⇒
![delim{[}{matrix{2}{2}{d c d c }}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_c22fb5710681e1aa104e9dd6d57dc348.png "delim{[}{matrix{2}{2}{d c d c }}{]}")
Problem 7.
Show [E] = [A][C] =
![delim{[}{matrix{2}{2}{
{B_11 D_11 + B_12 D_21} {B_11 D_12 + B_12 D_22}
{B_21 D_11 + B_22 D_21} {B_21 D_12 + B_22 D_22}
}}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_b81c1aebad28a4bc602980f50ff7317b.png "delim{[}{matrix{2}{2}{
{B_11 D_11 + B_12 D_21} {B_11 D_12 + B_12 D_22}
{B_21 D_11 + B_22 D_21} {B_21 D_12 + B_22 D_22}
}}{]}")
![[A] = delim{[}{matrix{2}{2}{B_11 B_12 B_21 B2_2 }}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_0be1bc63e9514a424fb92fb12e7b2990.png "[A] = delim{[}{matrix{2}{2}{B_11 B_12 B_21 B2_2 }}{]}")
,
![[C] = delim{[}{matrix{2}{2}{D_11 D_12 D_21 D2_2 }}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_47b1b821859ad45496293dca3fc6d8a2.png "[C] = delim{[}{matrix{2}{2}{D_11 D_12 D_21 D2_2 }}{]}")
,
![[E] = delim{[}{matrix{2}{2}{F_11 F_12 F_21 F2_2 }}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_f33abe4315b0d1a2fa5aa86c55c1717d.png "[E] = delim{[}{matrix{2}{2}{F_11 F_12 F_21 F2_2 }}{]}")
,
![B_11 = delim{[}{matrix{2}{2}{a_11 a_12 a_21 a2_2 }}{]},
B_12 = delim{[}{matrix{1}{2}{a_13 a_23}}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_84a85aac81767c1b992b997e0431a3c1.png "B_11 = delim{[}{matrix{2}{2}{a_11 a_12 a_21 a2_2 }}{]},
B_12 = delim{[}{matrix{1}{2}{a_13 a_23}}{]}")
![B_21 = delim{[}{matrix{2}{1}{a_31 a_32}}{]}, B_22 = delim{[}{matrix{1}{1}{a_33~}}{~]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_63ecaf9801b93f51aef5d2e7a2264daa.png "B_21 = delim{[}{matrix{2}{1}{a_31 a_32}}{]}, B_22 = delim{[}{matrix{1}{1}{a_33~}}{~]}")
.
![D_11 = delim{[}{matrix{2}{2}{c_11 c_12 c_21 c_22 }}{]}, D_12 = delim{[}{matrix{1}{2}{c_13 c_23}}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_5b64c56493c1d1badced19246ec246d0.png "D_11 = delim{[}{matrix{2}{2}{c_11 c_12 c_21 c_22 }}{]}, D_12 = delim{[}{matrix{1}{2}{c_13 c_23}}{]}")
![D_21 = delim{[}{matrix{2}{1}{c_31 c_32}}{]}, D_22 = delim{[}{matrix{1}{1}{c_33~}}{~]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_bb2900b07ffbb510e6ea3b90babbe0d9.png "D_21 = delim{[}{matrix{2}{1}{c_31 c_32}}{]}, D_22 = delim{[}{matrix{1}{1}{c_33~}}{~]}")
.
![F_11 = delim{[}{matrix{2}{2}{e_11 e_12 e_21 e_22 }}{]}, F_12 = delim{[}{matrix{1}{2}{e_13 e_23}}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_ec64eb9fc752c3a40419b39aa96fedc4.png "F_11 = delim{[}{matrix{2}{2}{e_11 e_12 e_21 e_22 }}{]}, F_12 = delim{[}{matrix{1}{2}{e_13 e_23}}{]}")
![F_21 = delim{[}{matrix{2}{1}{e_31 e_32}}{]}, F_22 = delim{[}{matrix{1}{1}{e_33~}}{~]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_3f0decd0f26c9abd3270b0dd12d9df32.png "F_21 = delim{[}{matrix{2}{1}{e_31 e_32}}{]}, F_22 = delim{[}{matrix{1}{1}{e_33~}}{~]}")
.
![[E] = delim{[}{matrix{2}{2}{
{B_11 D_11 + B_12 D_21} {B_11 D_12 + B_12 D_22}
{B_21 D_11 + B_22 D_21} {B_21 D_12 + B_22 D_22}
}}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_af0c9123ad9a0f589f92958f1963a221.png "[E] = delim{[}{matrix{2}{2}{
{B_11 D_11 + B_12 D_21} {B_11 D_12 + B_12 D_22}
{B_21 D_11 + B_22 D_21} {B_21 D_12 + B_22 D_22}
}}{]}")
=![[E] = delim{[}{matrix{2}{2}{
{delim{[}{matrix{2}{2}{a_11 a_12 a_21 a2_2 }}{]} delim{[}{matrix{2}{2}{c_11 c_12 c_21 c_22 ~}}{~]}
+ delim{[}{matrix{1}{2}{a_13 a_23}}{]} delim{[}{matrix{2}{1}{c_31 c_32~}}{~]}}
{delim{[}{matrix{2}{2}{a_11 a_12 a_21 a2_2 }}{]} delim{[}{matrix{1}{2}{c_13 c_23~}}{~]}
+ delim{[}{matrix{1}{2}{a_13 a_23}}{]} delim{[}{matrix{1}{1}{c_33~}}{~]}}
{delim{[}{matrix{2}{1}{a_31 a_32}}{]} delim{[}{matrix{2}{2}{c_11 c_12 c_21 c_22 ~}}{~]}
+ delim{[}{matrix{1}{1}{a_33}}{]} delim{[}{matrix{2}{1}{c_31 c_32~}}{~]}}
{delim{[}{matrix{2}{1}{a_31 a_32}}{]} delim{[}{matrix{1}{2}{c_13 c_23~}}{~]}
+ delim{[}{matrix{1}{1}{a_33}}{]} delim{[}{matrix{1}{1}{c_33~}}{~]}}
}}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_926.5_1162b7d945c8609c0fe2984a4455e647.png "[E] = delim{[}{matrix{2}{2}{
{delim{[}{matrix{2}{2}{a_11 a_12 a_21 a2_2 }}{]} delim{[}{matrix{2}{2}{c_11 c_12 c_21 c_22 ~}}{~]}
+ delim{[}{matrix{1}{2}{a_13 a_23}}{]} delim{[}{matrix{2}{1}{c_31 c_32~}}{~]}}
{delim{[}{matrix{2}{2}{a_11 a_12 a_21 a2_2 }}{]} delim{[}{matrix{1}{2}{c_13 c_23~}}{~]}
+ delim{[}{matrix{1}{2}{a_13 a_23}}{]} delim{[}{matrix{1}{1}{c_33~}}{~]}}
{delim{[}{matrix{2}{1}{a_31 a_32}}{]} delim{[}{matrix{2}{2}{c_11 c_12 c_21 c_22 ~}}{~]}
+ delim{[}{matrix{1}{1}{a_33}}{]} delim{[}{matrix{2}{1}{c_31 c_32~}}{~]}}
{delim{[}{matrix{2}{1}{a_31 a_32}}{]} delim{[}{matrix{1}{2}{c_13 c_23~}}{~]}
+ delim{[}{matrix{1}{1}{a_33}}{]} delim{[}{matrix{1}{1}{c_33~}}{~]}}
}}{]}")
Continuing to substitute the results of these multiplications leads directly to the long form of the series definition of this matrix product.
= ![[a_ik][c_kj]](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_980.5_fe75952042b3bef13015ab3eb5e4cc1c.png "[a_ik][c_kj]")
= [A][C].
Problem 10.
Prove |[A][B]| = |[A]||[B]|.
[A][B] =![delim{[}{matrix{2}{2}{ {a_11 b_11 + a_12 b_21} {a_11 b_12 + a_12 b_22} {a_21 b_11 + a_22 b_21} {a_21 b_12 +a_22 b_22 } }}{]}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_971.5_b7ac048948137f6c98a16b7274f42aa1.png "delim{[}{matrix{2}{2}{ {a_11 b_11 + a_12 b_21} {a_11 b_12 + a_12 b_22} {a_21 b_11 + a_22 b_21} {a_21 b_12 +a_22 b_22 } }}{]}")
|[A][B]| = 
=
=
=
=
=
Problem 11.
![delim{[}{ matrix{3}{3}{
3 1 {-lambda}
4 {-2} {-3}
{2 lambda} 4 lambda
} }{]}
delim{lbrace}{ matrix{3}{1}{ x_1 x_2 x_3 } }{rbrace}
=
delim{lbrace}{ matrix{3}{1}{ 0 0 0 } }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_951.5_2f2ff8f7ae3263923507b279c0a1199e.png "delim{[}{ matrix{3}{3}{
3 1 {-lambda}
4 {-2} {-3}
{2 lambda} 4 lambda
} }{]}
delim{lbrace}{ matrix{3}{1}{ x_1 x_2 x_3 } }{rbrace}
=
delim{lbrace}{ matrix{3}{1}{ 0 0 0 } }{rbrace}")
![delim{[}{ matrix{3}{3}{
3 1 {0}
4 {-2} {-3}
0 4 0
} }{]}
delim{lbrace}{ matrix{3}{1}{ x_1 x_2 x_3 } }{rbrace}
=
lambda
delim{[}{ matrix{3}{3}{
0 0 {1}
0 0 0
{-2} 0 {-1}
} }{]}
delim{lbrace}{ matrix{3}{1}{ x_1 x_2 x_3 } }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_951.5_6d603257797707366d80601eb8bcdc7f.png "delim{[}{ matrix{3}{3}{
3 1 {0}
4 {-2} {-3}
0 4 0
} }{]}
delim{lbrace}{ matrix{3}{1}{ x_1 x_2 x_3 } }{rbrace}
=
lambda
delim{[}{ matrix{3}{3}{
0 0 {1}
0 0 0
{-2} 0 {-1}
} }{]}
delim{lbrace}{ matrix{3}{1}{ x_1 x_2 x_3 } }{rbrace}")
is arbitrary and has no effect on λ.
![delim{[}{ matrix{2}{2}{
3 {0}
0 0
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}
=
lambda delim{[}{ matrix{2}{2}{
0 {1}
{-2 } {-1}
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_951.5_c7b1d53aefb3e8be1db562b717151875.png "delim{[}{ matrix{2}{2}{
3 {0}
0 0
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}
=
lambda delim{[}{ matrix{2}{2}{
0 {1}
{-2 } {-1}
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}")
![delim{[}{ matrix{2}{2}{
{-1/2} {-1/2}
1 0
} }{]}
delim{[}
{ matrix{2}{2}{
3 {0}
0 0
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}
=
lambda
delim{[}{ matrix{2}{2}{
{-1/2} {-1/2}
1 0
} }{]}
delim{[}{ matrix{2}{2}{
0 {1}
{-2} {-1}
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_951.5_0a309b49e04cdeb038e61093e589bf71.png "delim{[}{ matrix{2}{2}{
{-1/2} {-1/2}
1 0
} }{]}
delim{[}
{ matrix{2}{2}{
3 {0}
0 0
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}
=
lambda
delim{[}{ matrix{2}{2}{
{-1/2} {-1/2}
1 0
} }{]}
delim{[}{ matrix{2}{2}{
0 {1}
{-2} {-1}
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}")
⇒![delim{[}{ matrix{2}{2}{
{-3/2} {0}
3 0
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}
=
lambda
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_951.5_fe16d5326093427b4f05b828f34f99ae.png "delim{[}{ matrix{2}{2}{
{-3/2} {0}
3 0
} }{]}
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}
=
lambda
delim{lbrace}{ matrix{2}{1}{ x_1 x_3 } }{rbrace}")
Now it seems 
is messed up,
and all that is left is λ=-3/2 and the trivial solution λ=0.
For λ=-3/2, 
.
Going back to equation 2, 
=
⇒
⇒
