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TRANSCRIPT
Everything You Need to Know
About Linear Algebra in Twenty
Minutes
Benjamin G. Levine
This Lecture
• Definition of vector and matrix
• Definition of vector-vector, matrix-vector and matrix-matrix multiplication
• Eigenvalue problems
• Definition of commutator
• Definitions of terms related to square matrices
Vectors
1
2
...
M
v
vv
v
= =
v
Vector
Vectors
1
2
...
M
v
vv
v
= =
v
Vector Adjoint (or complex
transpose)
†
1 2 ...M
v v v∗ ∗ ∗ = v
Vectors
1
2
...
M
v
vv
v
= =
v
Vector Adjoint (or complex
transpose)
†
1 2 ...M
v v v∗ ∗ ∗ = v
Transpose
1 2 ...T
Mv v v = v
Vector-Vector Multiplication
1
2†
1 2
1
......
M
M i i
i
M
w
wv v v v w
w
∗ ∗ ∗ ∗
=
= =
∑v w
Inner Product
(AKA scalar
product; dot
product)
Vector-Vector Multiplication
1 1 1 1 2 1
2† 2 1 2 2 2
1 2
1 2
...
......
... ... ... ... ...
...
M
M
M
N N N N M
w w v w v w v
w w v w v w vv v v
w w v w v w v
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
= =
wv
Outer Product
Properties of Sets of Vectors
• Orthogonal set of vectors
• Orthonormal set of vectors
– are orthogonal
– have unit length (are normalized)
0=†u v
21≡ =†
v v v
Matrices
11 12 1
21 22 2
1 2
...
...
... ... ... ...
...
M
M
N N NM
A A A
A A A
A A A
=
A
Matrix (N x M)
Matrices
11 12 1
21 22 2
1 2
...
...
... ... ... ...
...
M
M
N N NM
A A A
A A A
A A A
=
A
Matrix (N x M)Adjoint (or complex
transpose)
11 21 1
† 12 22 2
1 2
...
...
... ... ... ...
...
N
N
M M NM
A A A
A A A
A A A
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
=
A
Matrix-Matrix Multiplication
11 12 1 11 12 111 12 1
21 22 2 21 22 221 22 2
1 2 1 21 2
... ......
... ......
... ... ... ... ... ... ... ...... ... ... ...
... ......
M LL
M LL
N N NM N N NLM M ML
A A A C C CB B B
A A A C C CB B B
A A A C C CB B B
= =
AB
Matrix-Matrix Multiplication
11 12 1 11 12 111 12 1
21 22 2 21 22 221 22 2
1 2 1 21 2
... ......
... ......
... ... ... ... ... ... ... ...... ... ... ...
... ......
M LL
M LL
N N NM N N NLM M ML
A A A C C CB B B
A A A C C CB B B
A A A C C CB B B
= =
AB
N x M M x L N x L
Matrix-Matrix Multiplication
11 12 1 11 12 111 12 1
21 22 2 21 22 221 22 2
1 2 1 21 2
... ......
... ......
... ... ... ... ... ... ... ...... ... ... ...
... ......
M LL
M LL
N N NM N N NLM M ML
A A A C C CB B B
A A A C C CB B B
A A A C C CB B B
= =
AB
1
M
ij ik kj
k
C A B=
=∑
N x M M x L N x L
Matrix-Matrix Multiplication
11 12 1 11 12 111 12 1
21 22 2 21 22 221 22 2
1 2 1 21 2
... ......
... ......
... ... ... ... ... ... ... ...... ... ... ...
... ......
M LL
M LL
N N NM N N NLM M ML
A A A C C CB B B
A A A C C CB B B
A A A C C CB B B
= =
AB
1
M
ij ik kj
k
C A B=
=∑
N x M M x L N x L
Matrix-Matrix Multiplication
11 12 1 11 12 111 12 1
21 22 2 21 22 221 22 2
1 2 1 21 2
... ......
... ......
... ... ... ... ... ... ... ...... ... ... ...
... ......
M LL
M LL
N N NM N N NLM M ML
A A A C C CB B B
A A A C C CB B B
A A A C C CB B B
= =
AB
1
M
ij ik kj
k
C A B=
=∑
N x M M x L N x L
Matrix-Matrix Multiplication
11 12 1 11 12 111 12 1
21 22 2 21 22 221 22 2
1 2 1 21 2
... ......
... ......
... ... ... ... ... ... ... ...... ... ... ...
... ......
M LL
M LL
N N NM N N NLM M ML
A A A C C CB B B
A A A C C CB B B
A A A C C CB B B
= =
AB
1
M
ij ik kj
k
C A B=
=∑
N x M M x L N x L
Matrix-Matrix Multiplication
• Properties of matrix-matrix multiplication
– Commutative property does not necessarily
hold
– Associative property holds
≠AB BA
( ) ( )=AB C A BC
Matrix-Matrix Multiplication
– Distributive property holds
– For scalar multiplication, the associative and
commutative properties hold
( )+ = +A B C AB AC
( ) ( ) ( )c c c= =AB A B A B
c c=A A
is a scalarc
Matrix-Vector Multiplication
11 12 1 11
21 22 2 22
1 2
...
...
... ... ... ... ......
...
M
M
N N NM NM
A A A dc
A A A dc
A A A dc
= =
Ac
Matrix-Vector Multiplication
1
M
i ij j
j
d A c=
=∑
11 12 1 11
21 22 2 22
1 2
...
...
... ... ... ... ......
...
M
M
N N NM NM
A A A dc
A A A dc
A A A dc
= =
Ac
Matrix-Vector Multiplication
1
M
i ij j
j
d A c=
=∑
N x M M N
11 12 1 11
21 22 2 22
1 2
...
...
... ... ... ... ......
...
M
M
N N NM NM
A A A dc
A A A dc
A A A dc
= =
Ac
Matrix-Vector Multiplication
11 12 1 11
21 22 2 22
1 2
...
...
... ... ... ... ......
...
M
M
N N NM NM
A A A dc
A A A dc
A A A dc
= =
Ac
1
M
i ij j
j
d A c=
=∑
N x M M N
Eigenvalue Problems
ω=Oc c
VectorScalarSquare
Matrix
Eigenvalue Problems
• The Schrodinger equation is an eigenvalue problem
ω=Oc c
E=HΨ Ψ
Commutator
[ , ] ≡ −A B AB BA
if [ , ] 0 then
and have the same eigenvectors
=
=
A B
AB BA
A B
Square Matrices
11 12 1
21 22 2
1 2
...
...
... ... ... ...
...
N
N
N N NN
A A A
A A A
A A A
=
A
Square Matrix
(N x N)
Square Matrices
11 12 1
21 22 2
1 2
...
...
... ... ... ...
...
N
N
N N NN
A A A
A A A
A A A
=
A
Square Matrix
(N x N)
Trace
1
trN
ii
i
A=
≡∑A
Determinant
11 12
11 22 12 21
21 22
A AA A A A
A A= = −A
Determinant
11 12
11 22 12 21
21 22
A AA A A A
A A= = −A
Determinant
11 12
11 22 12 21
21 22
A AA A A A
A A= = −A
Determinant
11 12
11 22 12 21
21 22
A AA A A A
A A= = −A
Most Important Property:
The interchange of any two rows or
columns of changes the sign of . AA
Types of Square Matrices
• Diagonal Matrix
– are the eigenvalues of
0 for ijA i j= ≠
iiA A
Types of Square Matrices
• Diagonal Matrix
– are the eigenvalues of
• Unit Matrix
–
0 for ijA i j= ≠
( ) ( ) ijij ijδ= =I 1
= =1A A1 A
iiA A
Types of Square Matrices
• Diagonal Matrix
– are the eigenvalues of
• Unit Matrix
–
• Inverse Matrix
0 for ijA i j= ≠
( ) ( ) ijij ijδ= =I 1
= =1A A1 A
1 1− −= =A A AA 1
iiA A
Types of Square Matrices
• Hermitian Matrix
– Real eigenvalues, orthogonal eigenvectors
or ij jiA A∗= =†
A A
Types of Square Matrices
• Hermitian Matrix
– Real eigenvalues, orthogonal eigenvectors
• Unitary Matrix
–
– Unitary transformation
• and have the same eigenvalues, same trace,
same determinant
– Preserves length of vector
or ij jiA A∗= =†
A A
=†U U 1
= †A U BU
A B
=-1 †U U
2 2=v Uv
Eigenvalue Problems
ω=Oc c
VectorScalarMatrix
Eigenvalue Problems
• Solving an eigenvalue problem is diagonalizing a matrix
ω=Oc c
Ω = †U OU
Eigenvalue Problems
• Solving an eigenvalue problem is diagonalizing a matrix
ω=Oc c
Ω = †U OU
Find unitary
matrix
Eigenvalue Problems
• Solving an eigenvalue problem is diagonalizing a matrix
ω=Oc c
Ω = †U OU
Find unitary
matrixDiagonal
Eigenvalue Problems
• Solving an eigenvalue problem is diagonalizing a matrix
ω=Oc c
Ω = †U OU
Diagonal elements give
us the eigenvalues
Eigenvalue Problems
• Solving an eigenvalue problem is diagonalizing a matrix
ω=Oc c
Ω = †U OU
Diagonal elements give
us the eigenvaluesRows and columns of
unitary matrix give us
eigenvectors