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5.3 Two–by–Two Matrices: Index Notation and Multiplication
5.3.1 Basis Vectors and Index Notation
Vectors
The vectors e1=10e2=01 are called
basis vectors of ℝ2.
Any arbitrary vector a∈ℝ2
is written as a linear combination
a=a1e1+a2e2=∑2i=1aiei
In this representation, sometimes Einstein’s summation convention is used: We write
a=∑2i=1aiei=aiei, omitting
the sum symbol in order to simplify the notation. The sum is automatically carried out over repeated indices. Here, the
index is i.
Matrices
The element Aij of a
matrix A is the entry
in its i-th row and its
j-th column. For two-by-two
matrices, this reads A11A21A12A22
Note: be very careful not to mix up the row and the column index!
Matrix operating on vector
The result of a linear mapping x→y=Ax
can be written in index form, too:
↔x=x1x2→Ax=y=y1y2yi=∑2j=1Aijxj
This means that the first and second components, y1
and y2, of
y=Ax are
given by
y1=∑2j=1A1jxjy2=∑2j=1A2jxj
Note that the index j runs over
the columns of the matrix A.
5.3.2 Multiplication of a Matrix with a Scalar
This is simple,
λacbd=λaλcλbλd
5.3.3 Matrix Multiplication: Definition
A matrix A moves
a vector x into a
new vector y=Ax.
This new vector can again be transformed into another vector
y by acting with another
matrix B on it: y=By=BAx. The
combined operation C=BA
transforms the original vector x
into y in
one single step. This matrix product is calculated according to
In general, the matrix product does not commmute, i.e.,
AB≠BA
This means that in contrast to real or complex numbers, the result of a multiplication of two matrices
A and
B depends on
the order of A
and B.
The commutator [AB]
of two matrices A
and B is
defined as [AB]=AB−BA
The commutator plays a central role in quantum mechanics, where classical variables like position
x and
momentum p
are replaced by operators(matrices) which in general do not commute, i.e., their commutator is
non–zero.
The abstract way to write a matrix multiplication with indices:
C=BA⇝Cij=∑2k=1BikAkj(=BikAkj in the summation convention).
To get the element in the ith
row and jth column of
the product BA, take the
scalar product of the ith
row-vector of B with
the j-th column
vector of A.
This looks complicated but it is not, it is just another formulation of our definition Eq.(5.28).
10
e2=
