The Chemistry Maths Book, Second Edition

548 Chapter 19The matrix eigenvalue problem

The canonical form

We have seen (equation (19.30)) that a symmetric matrix Ais reduced to diagonal

form by the similarity transformationX

T

AXwhere Xis the orthogonal matrix whose

columns are the orthonormal eigenvectors of A. BecauseXX

T

1 = 1 I(for orthogonal

matrix X), we can write (19.38) as

Q 1 = 1 x

T

(XX

T

)A(XX

T

)x

= 1 (x

T

X) (X

T

AX) (X

T

x)

= 1 y

T

Dy (19.40)

whereD 1 = 1 X

T

AXis the diagonal matrix of the eigenvalues of A, and

y 1 = 1 X

T

x (19.41)

is the vector obtained from xby the orthogonal transformationX

T

.

The quadratic form Qin the nvariablesx

1

, x

2

, x

3

, =, x

n

has been transformed into

an equivalent form in the nvariablesy

1

, y

2

, y

3

, =, y

n

that contains only pure square

terms:

(19.42)

This is the canonical formof Q, and the variables y

k

are the canonical variables.

3

EXAMPLE 19.14Transform the following quadratic form into canonical form:

Q 1 = 15 x

1

2

1 + 18 x

1

x

2

1 + 15 x

2

2

We have

and the orthonormal eigenvectors of the symmetric matrix Aare

xx

12

1

2

1

1

1

2

1

1

=

−

















=

















,

Qxx

x

x

=

















=

























()

12

1

2

54

45

xAx

T

Qyyyyy

k

n

kk nn

()y ==++++

=

∑

1

2

11

2

22

2

33

22

λλλλ λ

3

Cayley and Sylvester developed the theory of forms between 1854 and 1878. Sylvester claimed that he

discovered and developed the reduction of a quadratic form to canonical form at one sitting ‘with a decanter of

port wine to sustain nature’s flagging energies’. A general description of canonical forms was given by Camille

Jordan (1838–1922) in his Traité des substitutions et des équations algébriques(Treatise on substitutions and

algebraic equations) of 1871, in which he presented many of the modern concepts of group theory within the

context of groups of permutations (substitutions).