Page 18 - Advanced Linear Algebra

2 Advanced Linear Algebra

field are called scalars . We expect that the reader is familiar with the basic
-
properties of matrices, including matrix addition and multiplication.
The main diagonal of an d matrix is the sequence of entries
(
Á
(Á (Á Ã Á ( Á
Á
where ~ min ¸ Á ¹ .
Definition The transpose of ( C Á is the matrix ( ! defined by
!
²( ³ Á ~ ( Á
A matrix is symmetric if ( ~ ( ! and skew-symmetric if ( ! ~ c ( .
(
( ) . Then
Theorem 0.1 Properties of the transpose Let , ()  C Á
!!
1) ²( ³ ~ (
!
!
2) ²( b )³ ~ ( b ) !
!
)
3 ² (³ ~ ( ! for all  -
!
!
)
4 ²()³ ~ ) ( ! provided that the product () is defined
!
5)det²( ³ ~ det²(³ .
Partitioning and Matrix Multiplication
Let 4 be a matrix of size d . If ) ‹ ¸ Á à Á ¹ and * ‹ ¸ Á à Á ¹ , then

the submatrix 4´)Á *µ is the matrix obtained from 4 by keeping only the
rows with index in and the columns with index in . Thus, all other rows and
)
*
(
((
columns are discarded and 4´)Á *µ has size ) d *( .
Suppose that 4 C and 5 C Á Á . Let
1) F ~ ¸) Á ÃÁ) ¹ be a partition of ¸ Á ÃÁ ¹

2) G ~ ¸* Á ÃÁ* ¹ be a partition of ¸ Á ÃÁ ¹

3) H ~ ¸+ Á ÃÁ+ ¹ be a partition of ¸ Á ÃÁ ¹

(Partitions are defined formally later in this chapter.) Then it is a very useful fact
that matrix multiplication can be performed at the block level as well as at the
entry level. In particular, we have
´45µ´) Á + µ ~ 4´) Á * µ5´* Á + µ

*G

When the partitions in question contain only single-element blocks, this is
precisely the usual formula for matrix multiplication

~
´45µ Á 4 5 Á
Á
~