John Stillwell - Naive Lie Theory.pdf - Index of

1.2 Matrix representation of complex numbers 5
1.1.3 Also deduce, from Exercise 1.1.1, that
tan(θ + ϕ)=
tanθ + tanϕ
1 − tanθ tanϕ .
1.1.4 Using Exercise 1.1.3, or otherwise, write down the formula for tan(θ − ϕ),
and deduce that lines through O at angles θ and ϕ are perpendicular if and
only if tanθ = −1/tanϕ.
1.1.5 Write down the complex number z −θ and the inverse of the matrix for rotation
through θ, and verify that they correspond.
1.2 Matrix representation of complex numbers
A good way to see why the matrices R θ = ( )
cosθ −sinθ
sinθ cos θ behave the same
as the complex numbers z θ = cosθ + isinθ is to write R θ as the linear
combination
( ) ( )
1 0 0 −1
R θ = cosθ + sinθ
0 1 1 0
of the basis matrices
1 =
It is easily checked that
( ) 1 0
, i =
0 1
( ) 0 −1
.
1 0
1 2 = 1, 1i = i1 = i, i 2 = −1,
so the matrices 1 and i behave exactly the same as the complex numbers 1
and i.
In fact, the matrices
( ) a −b
= a1 + bi, where a,b ∈ R,
b a
behave exactly the same as the complex numbers a + bi under addition
and multiplication, so we can represent all complex numbers by 2 × 2 real
matrices, not just the complex numbers z θ that represent rotations. This
representation offers a “linear algebra explanation” of certain properties of
complex numbers, for example:
• The squared absolute value, |a+bi| 2 = a 2 +b 2 of the complex number
a + bi is the determinant of the corresponding matrix ( )
a −b
b a .