Graphics Reference
In-Depth Information
Definition.
If A = (a
ij
) is an n ¥ n matrix, then the
trace
of A, denoted by tr(A), is
defined by
n
Â
()
=
tr A
a
ii
.
i
=
1
C.3.2. Theorem.
The trace function satisfies the following properties:
(1) tr(aA + bB) = a tr A + b tr B.
(2) tr(AB) = tr(BA).
(3) tr(A) = tr(P
-1
AP) for any nonsingular matrix P.
Proof.
Parts (1) and (2) follow from some simple computations. Part (3) follows
from (2).
Definition.
Let A = (a
ij
) be an n ¥ m matrix. The rows of A can be thought of as
vectors in
R
m
. The
row rank
of A is the dimension of the subspace in
R
m
that these
vectors generate. Similarly, the columns of A can be thought of as vectors in
R
n
. The
column rank
of A is the dimension of the subspace in
R
n
that these vectors generate.
One can show that the row rank and column rank of a matrix are the same.
Definition.
The
rank
of a matrix is the common value of the row rank or column
rank. An n ¥ m matrix has
maximal rank
if its rank is the smaller of n or m.
C.3.3. Theorem.
The rank of a matrix is the dimension of its largest nonsingular
square submatrix. An n ¥ n matrix is nonsingular if and only if it has rank n.
Proof.
See [John67].
It is assumed that the reader is familiar how matrices are used to solve linear
systems of equations of the form
T
T
A
xb
=
,
in particular the method of Gauss elimination. (We need the transpose of the vectors
because in this topic vectors in
R
n
are treated as 1 ¥ n matrices.) We will not describe
the method here, but there is some terminology that one runs into when the method
is discussed, which we want to record for the sake of completeness. Recall that if
Gauss elimination, applied to a matrix A to get an upper triangular matrix U, does
not involve interchanging rows, then A can be written in the form
ALU
=
,
where L is a lower-triangular matrix and U is an upper-triangular matrix. This reduces
the system of equations above to the two systems
T
T
L
yb
=
and
