or, equivalently,
!Q I I Q I I
V VU@ @
%#(
$" ) ¨ £ § ( § % ¥ )
&'˜{

The choice of a method for solving linear systems will often depend on the structure of
the matrix . One of the most important properties of matrices is symmetry, because of
its impact on the eigenstructure of . A number of other classes of matrices also have
particular eigenstructures. The most important ones are listed below:
• H
Symmetric matrices: .
G
•
HG H
Hermitian matrices: .
I
• 3
Skewsymmetric matrices: .
• H
3
SkewHermitian matrices: .
I
• (H
‚ I
I X
Normal matrices: .
˜ i)11YGa`gp )H…
(
• 7
Nonnegative matrices: (similar de¬nition for nonpositive,
positive, and negative matrices).
• 1I 0
0
Unitary matrices: .
d
u
¡£ ¢ ¡ ¥ ¥ £ £ © ¢§
¡ £
§
©
0
It is worth noting that a unitary matrix is a matrix whose inverse is its transpose conjugate
0 , since
IP
t
0I0 I0 0 0
¤
©¦ §¥
¨¦
&
0 1I 0
0
A matrix such that is diagonal is often called orthogonal.
Some matrices have particular structures that are often convenient for computational
purposes. The following list, though incomplete, gives an idea of these special matrices
which play an important role in numerical analysis and scienti¬c computing applications.
• H… ‚
8
7
Diagonal matrices: for . Notation:
‘‘‘ C‚1)11‘E667C‚ 0 0 ‚kB ¤
(
C
• p… ‚
7
Upper triangular matrices: for .
• l p… ‚
7
Lower triangular matrices: for .
• — s s H… ‚
a
7
Upper bidiagonal matrices: for or .
• s s XH…z‚
a
7 G
3
Lower bidiagonal matrices: for or .
Q !
• (H…z‚
a`
8
7 P
3
Tridiagonal matrices: for any pair such that . Notation:
C 0 4 „T‚TV„T‚ 0 ‚kB ¤
1 ( #
" #
"
&
• H ‚
…
— # $"
%
!"
%"
8
7 3B
Banded matrices: only if , where and are two
#!
—D" —p
nonnegative integers. The number is called the bandwidth of .
!
%
• H… ‚
a`
— & 7
Upper Hessenberg matrices: for any pair such that . Lower
Hessenberg matrices can be de¬ned similarly.
•
Outer product matrices: , where both and are vectors.
I
V V
•
Permutation matrices: the columns of are a permutation of the columns of the
identity matrix.
• Block diagonal matrices: generalizes the diagonal matrix by replacing each diago
nal entry by a matrix. Notation:
C A‘‘ i)11 6 6 0 0 !(¤ '
( B
• Block tridiagonal matrices: generalizes the tridiagonal matrix by replacing each
nonzero entry by a square matrix. Notation:
C 0 " V 0 " B (¤ ' )
R1
( 4
&