Re: Matrix - What the Crazy Node is this ?
Posted: 28 Oct 2013, 06:20
the matrix is a spreadsheet, you can feel it when you go to work, when you go to church, when you pay your taxes
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M = M
i,j 0,3
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| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
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-1
M . M = I
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-1 | 1 |
M = | - |
| x |
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-1
I . I = I
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M' = M
ij ji
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| 0.707 -0.900 |
M = | |
| 0.866 0.707 |
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| 0.707 0.866 |
T = | |
| -0.900 0.707 |
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R = M + L
ij ij ij
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R = M + L
| A B C | | J K L |
| | | |
= | D E F | + | M N O |
| | | |
| G H I | | P Q R |
| A+J B+K C+L |
| |
= | D+M E+N F+O |
| |
| G+P H+Q I+R |
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R = M - L
ij ij ij
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R = M - L
| A B C | | J K L |
| | | |
= | D E F | - | M N O |
| | | |
| G H I | | P Q R |
| A-J B-K C-L |
| |
= | D-M E-N F-O |
| |
| G-P H-Q I-R |
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n
--
R = \ M x L
ij / ik kj
--
k=1
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| A B C D |
M = | E F G H |
| I J K L |
| M N O P |
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L = | Q R |
| S T |
| U V |
| W X |
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R = M x L
| A B C D | | Q R |
= | E F G H | x | S T |
| I J K L | | U V |
| M N O P | | W X |
| AQ+BS+CU+DW AR+BT+CV+DX |
= | EQ+FS+GU+HW ER+FT+GV+HX |
| IQ+JS+KU+LW IR+JT+KV+LX |
| MQ+NS+OU+PW MR+NT+OV+PX |
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-1
M
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0
M
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1
M
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2
M
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3
M
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2
M = M . M
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n
I = I
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| A B C D |
M = | E F G H |
| I J K L |
| M N O P |
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| x1 y1 z1 1|
V = | x2 y2 z2 1|
| x3 y3 z3 1|
| x4 y4 z4 1|
| x5 y5 z5 1|
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V . M = V'
| x1 y1 z1 1 | | A B C D | | x1.A+y1.E+z1.I+1.M x1.B+y1.F+z1.J+1.N x1.C+y1.G+z1.K+1.O ... |
| x2 y2 z2 1 | . | E F G H |= | x2.A+y2.E+z2.I+1.M x2.B+y2.F+z2.J+1.N x2.C+y2.G+z2.K+1.O ... |
| x3 y3 z3 1 | | I J K L | | x3.A+y3.E+z3.I+1.M x3.B+y3.F+z3.J+1.N x3.C+y3.G+z3.K+1.O ... |
| x4 y4 z4 1 | | M N O P | | x4.A+y4.E+z4.I+1.M x4.B+y4.F+z4.J+1.N x4.C+y4.G+z4.K+1.O ... |
| x5 y5 z5 1 | | x5.A+y5.E+z5.I+1.M x5.B+y5.F+z5.J+1.N x5.C+y5.G+z5.K+1.O ... |
That's if you are using column vectors, like most of the world does (e.g. most math, physics, and engineering references).Daniel Brassard wrote: If a 4x4 matrix is defined as follows:
| A B C D |
| E F G H |
| I J K L |
| M N O P |
Then the direction vector for each axis is as follows:
X-axis = [ A E I ]
Y-axis = [ B F J ]
Z-axis = [ C G K ]
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M = [ 1 ].
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-1
M = [ 1 / M[0][0] ]
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| A B |
M = | |
| C D |
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D = AD - CB
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|A B C|
|D E F|
|G H I|
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D = A (E.I-H.F) – B (D.I-G.F) + C (D.H-G.E)
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| 2 0 0 0 |
M = | 0 2 0 0 |
| 0 0 0.5 0 |
| 0 0 0 1 |