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**Extra resources for Schaum's Outlines of Linear Algebra (5th Edition)**

**Example text**

Thus, 2 pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 3 2 3 1= 3 1= 3 1= 3 1 1 1 6 pﬃﬃﬃ pﬃﬃﬃ 7 6 6 7 7 A ¼ 4 0 À1 15 and P¼6 0 À1= 2 1= 2 7 4 5 pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 2 À1 À1 2= 6 À1= 6 À1= 6 Complex Matrices: Hermitian and Unitary Matrices 2 3 ! 31. Find AH where (a) A ¼ 6 þ 7i 1 þ 8i À6 À i 5i Recall that AH ¼ AT , the conjugate tranpose of A. Thus, ! 32. Show that A ¼ 1 3 À 23 i À 23 i 2 3i À 13 À 23 i # is unitary. The rows of A form an orthonormal set: 1 2 2 1 2 2 1 4 4 þ ¼1 À i; i Á À i; i ¼ þ 3 3 3 3 3 3 9 9 9 1 2 2 2 1 2 2 4 2 4 À i; i Á À i; À À i ¼ iþ þ À iÀ ¼0 3 3 3 3 3 3 9 9 9 9 2 1 2 2 1 2 4 1 4 ¼1 þ À i; À À i Á À i; À À i ¼ þ 3 3 3 3 3 3 9 9 9 Thus, A is unitary.

Compute AB using block multiplication, where 2 3 2 1 2 1 1 2 3 4 5 4 A¼ 3 4 0 and B¼ 4 5 6 0 0 2 0 0 0 3 1 15 1 ! R S F and B ¼ , where E; F; G; R; S; T are the given blocks, and 01Â2 and 01Â3 01Â3 T 01Â2 G are zero matrices of the indicated sites. Hence, Here A ¼ E ER 01Â3 AB ¼ ES þ FT GT ! 36. Let M ¼ diagðA; B; CÞ, where A ¼ ! 3 2 3 ! 9 12 15 4 1 7 3 þ 4 5 0 5 ¼ 19 26 33 7 7 0 0 0 2 2 ! 1 3 2 . Find M 2 . , B ¼ ½5, C ¼ 5 7 4 Because M is block diagonal, square each block: ! 7 10 2 ; B2 ¼ ½25; A ¼ 15 22 16 C ¼ 40 2 !

Consider the following two block matrices: 2 1 61 6 A¼6 69 44 3 2 1 8 4 5 3 1 7 4 3 4 1 6 4 5 3 5 17 7 57 7 45 3 2 and 1 61 6 B¼6 69 44 3 2 1 8 4 5 3 1 7 4 3 4 1 6 4 5 3 5 17 7 57 7 45 3 The block matrix A is not a square block matrix, because the second and third diagonal blocks are not square. On the other hand, the block matrix B is a square block matrix. Block Diagonal Matrices Let M ¼ ½Aij be a square block matrix such that the nondiagonal blocks are all zero matrices; that is, Aij ¼ 0 when i 6¼ j.