19.4 Matrix diagonalization 545
EXAMPLE 19.11Diagonalization
The inverse of the matrix Xof the eigenvectors of Ain Example 19.10 is
so that
0 Exercises 25–28
Similarity transformations
Two square matrices Aand Bare called similar matricesor similarity transformsif
they are related by the similarity transformation
2B 1 = 1 C
− 1AC (19.29)
where C is a nonsingular matrix. Equation (19.28) is therefore a similarity
transformation that reduces Ato diagonal form. When Ais symmetric then, by theorem
(19.23), the eigenvectors of Aare orthonormal and Xin (19.28) is an orthogonal
matrix, withX
− 11 = 1 X
T. A symmetric matrix is therefore reduced to diagonal form
by the orthogonal transformation
D 1 = 1 X
TAX (19.30)
Two important invariance propertiesof similarity transforms follow from equations
(18.38) to (18.40) for the determinant and trace of a matrix product: ifB 1 = 1 C
− 1AC
then
det 1 B 1 = 1 det 1 A invariance of the determinant (19.31)
tr 1 B 1 = 1 tr 1 A invariance of the trace (19.32)
For example,
det 1 B 1 = 1 det 1 (C
− 1AC) 1 = 1 det 1 (ACC
− 1) 1 = 1 det 1 (AI) 1 = 1 det 1 A
XAX
−=
−
−−
−
−
−
−
110 1
411
311
211
11 4 5
1100
011
123
111
100
01
−
=
−
00
002
=D
X
−=
−
−−
−
110 1
411
311
2In his 1878 monograph on the theory of matrices, Frobenius defined similar matrices, discussed the
properties of orthogonal matrices and transformations, and showed the relationship between the algebras of matrices
and quaternions by determining four 2 1 × 1 2 matrices whose algebra is that of the quaternion quantities 1, i,j,k.