205 - Suppose A3 is to be computed for the 44 matrix A...

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1 Suppose A 3 is to be computed for the 4 × 4 matrix A below, which is “partitioned” as a 2 × 2 “block matrix”. Suppose the “block matrix” is treated as an ordinary matrix containing scalar entries and its power computed accordingly: It can be verified by direct computation of A 3 that the above result is also correct. Matrix partition and block matrices: [] , ij ij AA BB ⎡⎤ == ⎢⎥ ⎣⎦
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2 Compatible partitions for block matrix multiplication: column partition of the first matrix = row partition of the second matrix Block matrix multiplication with compatible partitions [] ik ij jk j CA BC A B ⎡⎤ == = = ⎣⎦ Example 11 12 21 22 A A A A = ⎢⎥ 11 12 B B B B = 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 :
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3 12 1 2 1 2 1 2 Then . p m nn n n AB A A A B B B ⎡⎤ = ⎣⎦ ⎢⎥ = ′′ == + + + ←→ ↑↑ ↓↓ bb b a a a b b aa a a ba b a b b " # "" # [] 11 22 Let and . np mn AB ⎤⎡ ⎥⎢ = =
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205 - Suppose A3 is to be computed for the 44 matrix A...

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