Lecture 20 Outline - (Matrix) space
•Linear transformations [Sections A.3]
•Changing Bases... [Sections A.4]
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Matrix Algebra: Linear Transformations, Changing Bases, and Hermitian Matrices, Study notes of Physics
The fundamental concepts of matrix algebra, including linear transformations, changing bases, and hermitian matrices. It explains how matrices represent linear transformations, the rules for matrix addition and multiplication, and the definition of transpose and hermitian matrices. The document also discusses unitary transformations and their significance in changing bases.
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Pre 2010
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Lecture 20 Outline - (Matrix) space
Linear transformations [Sections A.3]
Changing Bases... [Sections A.4]
Matrix elements
As on Adventures in Quantum Mechanics episode 19,
defined linear transformation
T
T
11
, T
21
,... , T
nn
T
e
j (^) 〉
T
1 j (^) | e 1 〉 + T 2 j
e
2 〉
T
nj
e
n
〉
i n
T
ij
(^) | e
i 〉
For orthonormal basis,
e
i |
ˆ
T
e
j (^) 〉
k n
T
kj
e i | e k 〉 = T
ij
So we have the following matrix representation:
T
T
T
11
T
12
T
1 n
T
21
T
22
T
2 n
T n 1 T n 2
T
nn
operators
linear transformations
matrices
Hermitian Matrices
Defining (complex) conjugate
T
∗
and hermitian conjugate:
T
†
T˜
∗ :
T
∗
T
(^11) ∗
T
(^12) ∗
T
∗
1 n
T
(^21) ∗
T
(^22) ∗
T
∗
2 n
T
n (^) ∗ 1
T
n (^) ∗
2
T
∗
nn
T
†
T
∗
11
T
(^21) ∗
T
∗
n
1
T
∗
12
T
(^22) ∗
T
∗
n
2
T
∗
1 n
T
(^2) ∗ n
T
∗
nn
Real if
T
∗
T
, (imaginary if
T
∗
T
Hermitian matrix:
T
†
T
, (anti-hermitian:
T
†
T
Note that
T˜
T
and
T
∗ ) ∗
and
( T † ) † = T.
inner product of two vectors:
〈 α | β 〉 = a † b.
Other
n
×
n
matrix defns
for matrix multiplication, in general,
ST
TS
re-introduce commutator:
[
ST
] =
ST
TS
Product: transpose
ST
T˜
S˜
and
ST
) † = S † T †
Unit matrix
I
has elements
I
ij
δ
ij
Inverse of matrix
T
−
1 T
TT
−
1
I
ie.
T
−
1
C˜
det(
T
exists when
det(
T
And
ST
−
1
T ) − 1 ( S ) − 1.
Changing Bases (no sic)
Switching to new basis vectors
f
n
〉 ...
we can write the old ones as
e
j (^) 〉
i n
S
ij
(^) | f
i 〉 :
| e 1 〉 = S
11
| f 1 〉 + S
21
f
2 〉
- S n 1 | f n 〉
| e 2 〉 = S
12
| f 1 〉 + S
22
f
2 〉
- S n 2 | f n 〉
| e n 〉 = S 1 n | f 1 〉 + S 2 n | f 2 〉 +
S
nn
f
n
〉
where
S
is a linear transformation...
a A vector in the new basis changes components:
f
Sa
e
ie.
a
i f
j n
S
ij
(^) a
j e
(OR
a e = S − 1 a f
).^
A linear transformation,
a
′ e = T e a e
, changes too:
consider
Sa
′ e
ST
e
a
e
ST
e S − 1 a f = T f
a^
f
Similar Base
A linear transformation in different basis,
T
f
ST
e S
−
1 :
these matrices are
similar
Definition: similar when
T
2
ST
1 S
−
1
If
S
is unitary, and the first matrix is orthonormal then
the second also is.
det(bases are diferent, can show thatFinally, while the linear transformations in different
T
f
)^
det(
T
e
)
and
Tr(
T
f
)^
Tr(
T
e
)
.