Definitions
3.1
•
A
matrix
is a rectangular array of number called the
entries
, or
elements
, of the matrix.
•
If A is an
m x n
matrix and B is an
n x r
matrix, then the
product
C = AB is an
m x r
matrix.
The (
i, j
) entry of the product is computed as follows: (see p. 139)
•
The
transpose
of an
m x n
matrix
A
is the
n x m
matrix
A
T
obtained by interchanging the
rows and columns of
A
.
That is, the
i
th column of
A
T
is the
i
th row of A for all
i
.
•
A square matrix A is
symmetric
if A
T
= A – that is, if A is equal to its own transpose.
3.3
•
If
A
is an
n x n
matrix, an
inverse
of
A
is an
n x n
matrix
A’
with the property that
AA
’ =
I
and
A’A
=
I
Where
I
=
I
n
is the
n x n
identity matrix.
If such an
A’
exists, then
A
is called
invertible
.
•
An
elementary matrix
is any matrix that can be obtained by performing an elementary
row operation on an identity matrix.
3.4
•
Let A be a square matrix.
A factorization of A as A = LU, where L is unit lower
triangular and U is upper triangular, is called an
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 Spring '08
 Yeap
 Linear Algebra, Algebra

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