6.2 Mathematical Intermezzo 155
- Upper triangular ifai j= 0 fori>j, which for a 4 × 4 matrix is of the form
a 11 a 12 a 13 a 14
0 a 22 a 23 a 24
0 0 a 33 a 34
0 0 0 ann
- Lower triangular ifai j= 0 fori<j
a 11 0 0 0
a 21 a 22 0 0
a 31 a 32 a 33 0
a 41 a 42 a 43 a 44
- Upper Hessenberg ifai j= 0 fori>j+ 1 , which is similar to a upper triangular except that
it has non-zero elements for the first subdiagonal row
a 11 a 12 a 13 a 14
a 21 a 22 a 23 a 24
0 a 32 a 33 a 34
0 0 a 43 a 44
- Lower Hessenberg ifai j= 0 fori<j+ 1
a 11 a 12 0 0
a 21 a 22 a 23 0
a 31 a 32 a 33 a 34
a 41 a 42 a 43 a 44
- Tridiagonal ifai j= 0 for|i−j|> 1
a 11 a 12 0 0
a 21 a 22 a 23 0
0 a 32 a 33 a 34
0 0 a 43 a 44
There are many more examples, such as lower banded with bandwidthpforai j= 0 fori>j+p,
upper banded with bandwidthpforai j= 0 fori<j+p, block upper triangular, block lower
triangular etc.
For a realn×nmatrixAthe following properties are all equivalent
- If the inverse ofAexists,Ais nonsingular.
- The equationAx= 0 impliesx= 0.
- The rows ofAform a basis ofRn.
- The columns ofAform a basis ofRn.
5.Ais a product of elementary matrices.
- 0 is not an eigenvalue ofA.
The basic matrix operations that we will deal with are addition and subtraction
A=B±C=⇒ai j=bi j±ci j, (6.2)
scalar-matrix multiplication
A=γB=⇒ai j=γbi j,
vector-matrix multiplication
