8.1 Complex Numbers
8.2 Conjugates and Division of
Complex Numbers
8.3 Polar Form and DeMoivre’s
Theorem
8.4 Complex Vector Spaces and
Inner Products
8.5 Unitary and Hermitian
Matrices
COMPLEX VECTOR
SPACES
8
469
harles Hermite was born on Christmas
Eve in Dieuze, France––the sixth of seven
children. His father was a cloth merchant who
had studied engineering. Although much of
his early education occurred at home, he went
to Paris to study when he was eighteen.
At the age of 20, Hermite published a paper
titled
Considerations on the Algebraic
Solution of the Equation of the Fifth Degree.
(Later in his life, Hermite showed how to
solve a general fifth-degree equation by means
of elliptic modular functions––such equations
cannot be solved generally by algebraic
means.) That same year, Hermite was
admitted to the École Polytechnique, but he
was dismissed from the school after one year.
While at the Polytechnique, Hermite became
acquainted with Joseph Liouville, who in turn
introduced Hermite to Carl Jacobi. Hermite’s
correspondence with Jacobi reveals Hermite’s
early understanding of abstract mathematics.
In 1848, Hermite gained a position at the
Polytechnique (the same institution that had
dismissed him five years earlier). With this ap-
pointment, Hermite’s career finally began to
take shape. In 1856 he was appointed to the
Academy of Sciences, in 1869 to the École
Normale, and finally in 1870 to a professor-
ship at the Sorbonne, a position he held until
his retirement in 1897.
Hermite is best known for his work with
elliptic functions, for his proof that
e
is a tran-
scendental number, and for his introduction of
what are today called Hermite polynomials.
Hermite was the first to use the term
orthogo-
nal matrices
, and was the first to prove that if
a matrix is equal to its own conjugate
transpose, then its eigenvalues must be real.
8.1 COMPLEX NUMBERS
Thus far in the text, the scalar quantities used have been
real numbers
. In this chapter, you
will expand the set of scalars to include
complex numbers.
In algebra it is often needed to solve quadratic equations such as
The
general quadratic equation is
and its solutions are
where the quantity in the radical,
is called the
discriminant.
If
then the solutions are ordinary real numbers. But what can you conclude about the solutions
of a quadratic equation whose discriminant is negative? For example, the equation
b
2
2
4
ac
≥
0,
b
2
2
4
ac
,
x
5
2
b
6
!
b
2
2
4
ac
2
a
ax
2
1
bx
1
c
5
0
x
2
2
3
x
1
2
5
0.
Charles
Hermite
1822–1901
C