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Review sheet for Math 415.01 Midterm I
•
A
matrix
consists of a rectangular array of numbers, or elements, arranged in
m
rows and
n
columns.
The element in the
i
th row and
j
th column is designated
a
ij
and the matrix can be written as (
a
ij
).
•
We associate with each matrix
A
several other matrices:
–
the transpose of the matrix
A
T
= (
a
ij
)
T
= (
a
ji
), wherein we interchange rows and columns,
–
the complex conjugate,
A
, wherein we replace each element by
its
complex conjugate, and
–
the adjoint,
A
*
=
A
T
, wherein we do both of the above.
•
One adds two matrices or multiples a matrix by a
scalar
(i.e. a complex number) in the obvious
term-by-term manner. If
A
= (
a
ij
) is an
m
×
n
matrix and
B
= (
b
k‘
) is an
n
×
r
matrix, then
AB
= (
∑
n
s
=1
a
is
b
sj
). That is to say, each element of the product matrix is the dot product of a row
of the ﬁrst factor and a column of the second
•
The dot product of two vectors, usually written (
x
,
y
) or
<
x
,
y
>
, can be written in terms of matrix
multiplication as
x
T
y
(we consider vectors as matrices with one column).
•
The identity matrix,

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