LU-factorization.pdf - 1 Matrix Operations 1.1 1.1.1 Operations Multiplying two matrices(3 Block form   A1 A2 B1 A3 A4 B3 B2 B4   = A1 B 1 A2 B 3

LU-factorization.pdf - 1 Matrix Operations 1.1 1.1.1...

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1 Matrix Operations 1.1 Operations 1.1.1 Multiplying two matrices [...] (3) Block form: A 1 A 2 A 3 A 4 B 1 B 2 B 3 B 4 = A 1 B 1 + A 2 B 3 A 1 B 2 + A 2 B 4 A 3 B 1 + A 4 B 3 A 3 B 2 + A 4 B 4 block row times block column! 1.1.2 Matrix Operation Laws Matrix multiplication is associative : ( AB ) C = A ( BC ) Matrix operations are distributive : A ( B + C ) = AB + AC ( B + C ) D = BD + CD Matrix operations are usually NOT commutative : EF 6 = FE Example of special matrix that is commutative below. 1.1.3 Special Matrices Part 1 Identity matrix I is a square n × n matrix: I = 1 0 0 · · · 0 0 0 1 0 · · · 0 0 0 0 1 · · · 0 0 : : : : : : 0 0 0 · · · 1 0 0 0 0 · · · 0 1 with IA = A and AI = A 1
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2 Cost of elimination Consider an n × n matrix: a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n a 31 a 32 · · · a 3 n : : : : a n 1 a n 2 · · · a nn pivot a 11 -----------→ n ( n - 1) operations a 11 a 12 · · · a 1 n 0 a 0 22 · · · a 0 2 n 0 a 0 32 · · · a 0 3 n : : : : 0 a 0 n 2 · · · a 0 nn pivot a 0 22 --------------→ ( n - 1)( n - 2) operations a 11 a 12 · · · a 1 n 0 a 0 22 · · · a 0 2 n 0 0 · · · a 00 3 n : : : : 0 0 · · · a 00 nn etc. Total cost for left-hand-side elimination (matrix A only): Cost = n ( n - 1) + ( n - 1)( n - 2) + · · · + 1 · 0 = n 2 - n + ( n - 1) 2 - ( n - 1) + · · · + 1 2 - 1 = (1 2 + · · · + ( n - 1) 2 + n 2 ) - (1 +
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  • Fall '19
  • Yulia Peet

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