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Lecture

Block matrices II, LU decomposition.pdf

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Department
Statistics
Course
STAT312
Professor
Douglas Wiens
Semester
Fall

Description
61 11. Block matrices II, LU decomposition • The inverse of a non-singular block matrix can be computed by à ! 1 P S R Q 1 ³ 1 ´ 1 ³ P S·´ P SQ R 1 1 = ³ ´ Q RP S Q RP 1S 1 · ³ ´ 1 Q RP 1S RP 1 This is established merely by multiplying the orig- inal block matrix by its alleged inverse, and verify- ing that the product is indeed the identity matrix. The previous identity is used here. This is a very useful result if onP or Q is much smaller than the other, which in turn has special struc- ture which makes it easy to invert. • Example: ³ ´ 1 110 I + 110 = I 1 + 62 so that à ! 1 à ¡ ¢ ! I 1 I+ 110 1 1 ( + 1) 1 0 1 = 1 0 ( + 1) 1 ( + 1) à ! 1 ( + 1) I 11 0 1 = + 1 10 1 • Another consequence of (10.1) isA is an × matrix and B is a × matrix ( ), then the eigenvalues oBA are those oAB together with zeros. Proof: Using the two formulas for the determi- nant of a block matrix, we obtain à ! I A ¯ 1 ¯ det B I = | I |¯ AB ¯ ¯ ¯ = ¯ 1AB ¯ = |I AB | and = |I || I BA | = |I BA | Thus, if and are the characteristic poly- nomials, we have ( ) ( ) = ( ) 63 and the roots of the characteristic equa( ) = 0 foBA are those of the characteristic equation ( ) = 0 forAB together with 0’s - the matriceAB and BA have the same non- zero eigenvalues; i= then alleigenvalues
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