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|00000000| 5c 73 65 63 74 69 6f 6e | 7b 62 61 6c 61 6e 63 65 |\section|{balance|
|00000010| 20 2d 2d 2d 20 4d 61 74 | 72 69 78 20 42 61 6c 61 | --- Mat|rix Bala|
|00000020| 6e 63 69 6e 67 7d 0d 5c | 6c 61 62 65 6c 7b 66 6e |ncing}.\|label{fn|
|00000030| 3a 62 61 6c 61 6e 63 65 | 7d 0d 5c 69 6e 64 65 78 |:balance|}.\index|
|00000040| 7b 62 61 6c 61 6e 63 65 | 28 29 7d 0d 5c 69 6e 64 |{balance|()}.\ind|
|00000050| 65 78 7b 6d 61 74 72 69 | 78 21 62 61 6c 61 6e 63 |ex{matri|x!balanc|
|00000060| 65 7d 0d 5c 62 65 67 69 | 6e 7b 72 61 69 6c 7d 0d |e}.\begi|n{rail}.|
|00000070| 42 61 6c 61 6e 63 65 20 | 3a 20 27 62 61 6c 61 6e |Balance |: 'balan|
|00000080| 63 65 27 20 27 28 27 20 | 4d 61 74 72 69 78 20 27 |ce' '(' |Matrix '|
|00000090| 29 27 20 3b 0d 5c 65 6e | 64 7b 72 61 69 6c 7d 0d |)' ;.\en|d{rail}.|
|000000a0| 0d 5c 76 65 72 62 2b 62 | 61 6c 61 6e 63 65 2b 20 |.\verb+b|alance+ |
|000000b0| 74 61 6b 65 73 20 61 20 | 73 71 75 61 72 65 20 61 |takes a |square a|
|000000c0| 72 67 75 6d 65 6e 74 20 | 6d 61 74 72 69 78 20 61 |rgument |matrix a|
|000000d0| 6e 64 20 61 74 74 65 6d | 70 74 73 20 74 6f 0d 62 |nd attem|pts to.b|
|000000e0| 61 6c 61 6e 63 65 20 74 | 68 65 20 69 6e 70 75 74 |alance t|he input|
|000000f0| 20 6d 61 74 72 69 78 20 | 73 6f 20 74 68 61 74 20 | matrix |so that |
|00000100| 74 68 65 20 72 6f 77 20 | 61 6e 64 20 63 6f 6c 75 |the row |and colu|
|00000110| 6d 6e 20 6e 6f 72 6d 73 | 20 61 72 65 0d 61 70 70 |mn norms| are.app|
|00000120| 72 6f 78 69 6d 61 74 65 | 6c 79 20 65 71 75 61 6c |roximate|ly equal|
|00000130| 2e 20 49 74 20 72 65 74 | 75 72 6e 73 20 61 20 6c |. It ret|urns a l|
|00000140| 69 73 74 20 77 69 74 68 | 20 65 6c 65 6d 65 6e 74 |ist with| element|
|00000150| 73 20 5c 76 65 72 62 2b | 74 2b 0d 61 6e 64 20 5c |s \verb+|t+.and \|
|00000160| 76 65 72 62 2b 41 62 2b | 2c 20 73 75 63 68 20 74 |verb+Ab+|, such t|
|00000170| 68 61 74 20 66 6f 72 20 | 61 6e 20 61 72 67 75 6d |hat for |an argum|
|00000180| 65 6e 74 20 24 41 24 3a | 0d 0d 5c 5b 20 41 62 20 |ent $A$:|..\[ Ab |
|00000190| 3d 20 74 5e 7b 2d 31 7d | 20 41 20 74 20 5c 5d 0d |= t^{-1}| A t \].|
|000001a0| 0d 24 41 62 24 20 69 73 | 20 74 68 65 20 62 61 6c |.$Ab$ is| the bal|
|000001b0| 61 6e 63 65 64 20 6d 61 | 74 72 69 78 2c 20 61 6e |anced ma|trix, an|
|000001c0| 64 20 24 74 24 20 69 73 | 20 72 65 66 65 72 72 65 |d $t$ is| referre|
|000001d0| 64 20 74 6f 20 61 73 20 | 74 68 65 0d 74 72 61 6e |d to as |the.tran|
|000001e0| 73 66 6f 72 6d 61 74 69 | 6f 6e 20 6d 61 74 72 69 |sformati|on matri|
|000001f0| 78 2e 0d 0d 5c 70 61 72 | 61 67 72 61 70 68 7b 45 |x...\par|agraph{E|
|00000200| 78 61 6d 70 6c 65 7d 0d | 5c 69 6e 64 65 78 7b 65 |xample}.|\index{e|
|00000210| 78 61 6d 70 6c 65 20 6f | 66 20 62 61 6c 61 6e 63 |xample o|f balanc|
|00000220| 65 28 29 7d 0d 5c 69 6e | 64 65 78 7b 62 61 6c 61 |e()}.\in|dex{bala|
|00000230| 6e 63 65 28 29 21 65 78 | 61 6d 70 6c 65 7d 0d 5c |nce()!ex|ample}.\|
|00000240| 62 65 67 69 6e 7b 76 65 | 72 62 61 74 69 6d 7d 0d |begin{ve|rbatim}.|
|00000250| 3e 20 41 20 3d 20 5b 20 | 31 2c 20 31 30 2c 20 31 |> A = [ |1, 10, 1|
|00000260| 30 30 3b 20 32 2c 2d 32 | 30 2c 32 30 30 3b 20 2d |00; 2,-2|0,200; -|
|00000270| 31 30 30 2c 20 33 2c 20 | 2d 33 30 5d 0d 20 41 20 |100, 3, |-30]. A |
|00000280| 3d 0d 20 20 20 20 20 20 | 20 20 31 20 20 20 20 20 |=.      |  1     |
|00000290| 20 20 20 20 31 30 20 20 | 20 20 20 20 20 20 31 30 |    10  |      10|
|000002a0| 30 20 20 0d 20 20 20 20 | 20 20 20 20 32 20 20 20 |0  .    |    2   |
|000002b0| 20 20 20 20 20 2d 32 30 | 20 20 20 20 20 20 20 20 |     -20|        |
|000002c0| 32 30 30 20 20 0d 20 20 | 20 20 20 2d 31 30 30 20 |200  .  |   -100 |
|000002d0| 20 20 20 20 20 20 20 20 | 20 33 20 20 20 20 20 20 |        | 3      |
|000002e0| 20 20 2d 33 30 20 20 0d | 3e 20 6e 6f 72 6d 28 41 |  -30  .|> norm(A|
|000002f0| 29 0d 20 20 20 20 20 20 | 33 33 30 0d 3e 20 6e 6f |).      |330.> no|
|00000300| 72 6d 28 41 27 29 0d 20 | 20 20 20 20 20 32 32 32 |rm(A'). |     222|
|00000310| 0d 3e 20 72 65 73 20 3d | 20 62 61 6c 61 6e 63 65 |.> res =| balance|
|00000320| 28 41 29 0d 20 72 65 73 | 20 3d 0d 20 20 20 61 62 |(A). res| =.   ab|
|00000330| 20 20 20 20 20 20 20 20 | 20 20 20 74 20 20 20 20 |        |   t    |
|00000340| 20 20 20 20 20 20 20 20 | 0d 3e 20 72 65 73 2e 61 |        |.> res.a|
|00000350| 62 0d 20 41 62 20 3d 0d | 20 20 20 20 20 20 20 20 |b. Ab =.|        |
|00000360| 31 20 20 20 20 20 20 20 | 20 31 30 30 20 20 20 20 |1       | 100    |
|00000370| 20 20 20 20 31 30 30 20 | 20 0d 20 20 20 20 20 20 |    100 | .      |
|00000380| 30 2e 32 20 20 20 20 20 | 20 20 20 2d 32 30 20 20 |0.2     |   -20  |
|00000390| 20 20 20 20 20 20 20 32 | 30 20 20 0d 20 20 20 20 |       2|0  .    |
|000003a0| 20 2d 31 30 30 20 20 20 | 20 20 20 20 20 20 33 30 | -100   |      30|
|000003b0| 20 20 20 20 20 20 20 20 | 2d 33 30 20 20 0d 3e 20 |        |-30  .> |
|000003c0| 6e 6f 72 6d 28 72 65 73 | 2e 61 62 29 0d 20 20 20 |norm(res|.ab).   |
|000003d0| 20 20 20 31 35 30 0d 3e | 20 6e 6f 72 6d 28 72 65 |   150.>| norm(re|
|000003e0| 73 2e 61 62 27 29 0d 20 | 20 20 20 20 20 32 30 31 |s.ab'). |     201|
|000003f0| 0d 5c 65 6e 64 7b 76 65 | 72 62 61 74 69 6d 7d 0d |.\end{ve|rbatim}.|
|00000400| 0d 53 65 65 20 61 6c 73 | 6f 3a 20 50 61 67 65 7e |.See als|o: Page~|
|00000410| 5c 70 61 67 65 72 65 66 | 7b 66 6e 3a 6e 6f 72 6d |\pageref|{fn:norm|
|00000420| 7d 20 5c 76 65 72 62 2b | 6e 6f 72 6d 28 29 2b 2e |} \verb+|norm()+.|
|00000430| 0d 0d 5c 6e 65 77 70 61 | 67 65 0d 0d 0d 0d 0d 0d |..\newpa|ge......|
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