Lecture 2
Matrix Operations
•transpose, sum & difference, scalar multiplication
•matrix multiplication, matrix-vector product
•matrix inverse
2–1
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Lecture 2 Matrix Operations, Schemes and Mind Maps of Linear Algebra
Lecture 2. Matrix Operations. • transpose, sum & difference, scalar multiplication. • matrix multiplication, matrix-vector product. • matrix inverse. 2–1 ...
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2022/2023
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Lecture 2
Matrix Operations
•^
transpose, sum & difference, scalar multiplication
-^
matrix multiplication, matrix-vector product
-^
matrix inverse
Matrix transpose
transpose
of
m
×
n
matrix
A
, denoted
A
T^
or
A
′, is
n
×
m
matrix with
( A
)T
ij^
A
ji
rows and columns of
A
are transposed in
A
T
example:
^
^ T
[^
]^.
•^
transpose converts row vectors to column vectors, vice versa
-^
( A
)T
T^
A
Matrix Operations
Properties of matrix addition
•^
commutative:
A
B
B
A
•^
associative:
A
B
C
A
B
C
), so we can write as
A
B
C
•^
A
A
A
;^ A
A
•^
(A
B
T)
A
T^
B
T
Matrix Operations
Scalar multiplication
we can multiply a number (a.k.a.
scalar
) by a matrix by multiplying every
entry of the matrix by the scalarthis is denoted by juxtaposition or
·, with the scalar on the left:
^
^
^
(sometimes you see scalar multiplication with the scalar on the right)^ •
α^
β
)A
αA
βA
;^ (
αβ
)A
α)(
βA
•^
α(
A
B
αA
αB
•^
·^ A
;^1
·^
A
A
Matrix Operations
Examples
example 1:
[^
] [
]
[^
]
for example, to get
,^1
entry of product:
C
11
A
11
B
11
A
12
B
21
example 2:
[^
] [
]
[^
]
these examples illustrate that matrix multiplication is not (in general)commutative: we don’t (always) have
AB
BA
Matrix Operations
Properties of matrix multiplication
•^
0 A
,^ A
(here
can be scalar, or a compatible matrix)
•^
IA
A
,^ AI
A
•^
(AB
)C
A
(BC
), so we can write as
ABC
•^
α(
AB
αA
)B
, where
α
is a scalar
•^
A
(B
C
AB
AC
,^ (
A
B
)C
AC
BC
•^
(AB
T)
B
T^ A
T
Matrix Operations
Inner product
if^
v^
is a row
n
-vector and
w
is a column
n
-vector, then
vw
makes sense,
and has size
×
,^ i.e.
, is a scalar:
vw
v
w 1
vn
w n
if^
x^
and
y
are
n
-vectors,
x
T^ y
is a scalar called
inner product
or
dot
product
of
x
,^ y
, and denoted
x, y
〉^
or
x
·^
y:
〈x, y
〉^ =
x
T^ y
x
y 1
x
yn n
(the symbol
·^
can be ambiguous — it can mean dot product, or ordinary
matrix product) Matrix Operations
Matrix powers
if matrix
A
is square, then product
AA
makes sense, and is denoted
A
2
more generally,
k
copies of
A
multiplied together gives
A
k:
A
k^
A A
A
︸^
k
by convention we set
A
I
(non-integer powers like
A
1 /
2 are tricky — that’s an advanced topic)
we have
A
kA
l^ =
A
k+
l
Matrix Operations
Examples
example 1:
[^
]−
1
[^
]
(you should check this!)
example 2:
[^
]
does not have an inverse; let’s see why:
[^
a^
b c^
d
] [
]
[^
a^
b^
a^
b
c^ −
d^
c^ + 2
d
]
[^
]
... but you can’t have
a
b^
and
a^
b^
Matrix Operations
Properties of inverse
•^
( A
−^1
A
,^ i.e.
, inverse of inverse is original matrix
(assuming
A
is invertible)
•^
(AB
1
B
−^1
A
−^1
(assuming
A
,^ B
are invertible)
•^
( A
)T
−^1
A
−^1
)T
(assuming
A
is invertible)
•^
−I
I
•^
(αA
/α
)A
−^1
(assuming
A
invertible,
α
•^
if^
y^
Ax
, where
x
R
n^
and
A
is invertible, then
x
A
−^1
y:
A
−^1
y^
A
−^1
Ax
Ix
x
Matrix Operations