Matrix Algebra
properties of invertible matrix
condition number of matrix:
big condition number means the matrix is in ill-condition and it maybe not invertible
Here are some statements either all true or all false:
a.A is an invertible matrix
b.A is row equavalent to
identity matrix
c.A has n pivot positions
d.The equation
has only the trivial solution
e.The columns
from a linear identity set.
f. The linear transform
is one to one
g.The equation
has at least one solution for each
in
h.The columns of
span
i.The linear transformation
maps
onto
j.There is an
matrix
such that
or
i.
is an invertible matrix
Partitioned Matrices
A matrix of the form is said to be block upper triangular. Assume that is , is and is invertible. Then
Matrix Factorizations
LU Factorization
Then the solution of
could be get through
and
Subspaces of
The subspaces of
is any set
in
that has three propoerties:
a.The zero vector in
.
b.For each
and
in
, the sum
is in
c.For each
in
and each scalar
, the vector
is in H
The Null space of a matrix A is the set Nul A of all solutions of the homogeneous equation
The Null space of a
matrix A is subspace of
Basis: A basis for a subspace H of
is a linearly independent set in H that span H
The pivot columns of a matrix
form a basis for the column space of
.
Demension and Rank
Demension of a nonzero subspace
, denoted by dim
, is the number of vectors in any basis for
. The demension of the zero subspace {0} is definded to be zero.
The rank of a matrix
, denoted by rank
, is the demension of the column space of
.
If a matrix
has n columns, then
Rank and the Invertible matrix theorem
Let
be
matrix. Then the following statements are each equivalent to the statement that
is an invertible matrix:
m. the columns of
form a basis of
n. Col A=
o. dim Col A=n
p. rank A=n
q. Nul A={0}
r. dim Nul A=0