λ 1 = 1, X 1 =col[

√

3 ,1] and λ 2 = 2, X 2 =col[1,−

√

3].

Let Q= [X 1 , X 2 ] =

[√

3 1

1 −

√

3

]

=

[

x 11 x 12

x 21 x 22

]

denote the matrix containing the eigen-

vectors of Afor its column vectors. By definition the eigenvalues and eigenvectors

satisfy the equations

AX 1 =λ 1 X 1 and AX 2 =λ 2 X 2 ,

and these equations can be expressed using the above notation as

[ 5

4 −

√ 3

4

−

√ 3

4

7

4

][

x 11

x 21

]

=

[

λ 1 x 11

λ 1 x 21

]

and

[ 5

4 −

√ 3

4

−

√ 3

4

7

4

][

x 12

x 22

]

=

[

λ 2 x 12

λ 2 x 22

]

.

These two sets of linear equations can be represented by the single matrix equation

[

(^54) −√ 43
−
√ 3
4
7
4
][
x 11 x 12
x 21 x 22
]

[
x 11 x 12
x 21 x 22
][
λ 1 0
0 λ 2
]

. (10 .33)

The matrix whose column vectors are n-linearly independent eigenvectors of Ais

called the modal matrix associated with A. Here Qis the modal matrix of A. Denote

the diagonal matrix having the eigenvalues of Afor the elements on the diagonal as

D=diag (λ 1 , λ 2 ),then the equation (10.33) can be written as

AQ =QD (10 .34)

Left multiplication by Q−^1 gives Q−^1 AQ =D, where

Q=

[√

3 1

1 −

√

3

]

, Q −^1 =^1

4

[√

3 1

1 −

√

3

]

, D =diag (1 ,2)

This example illustrates that the modal matrix can be used to reduce a given matrix

to a diagonal form.

Changes of variables of the form D=Q−^1 AQ , for the proper choice of the ma-

trix Q, is a transformation often used to produce diagonal matrices in a variety of

applications.