linear-guest

15.1 Review Problems 281
To diagonalize a real symmetric matrix, begin by building an orthogonal
matrix from an orthonormal basis of eigenvectors, as in the example below.
Example 142 The symmetric matrix
( ) 2 1
M = ,
1 2
( ( )
1 1
has eigenvalues 3 and 1 with eigenvectors and respectively. After normalizing
these eigenvectors, we build the orthogonal
1)
−1
matrix:
Notice that P T P = I. Then:
MP =
P =
( 3 √2 1 √2
3 √
2
)
√−1
2
( 1 √2 1 √2
1 √
2
=
)
√−1
2
.
( 1 √2 1 √2
1 √
2
) ( )
3 0
.
√−1
2
0 1
In short, MP = P D, so D = P T MP . Then D is the diagonalized form of M
and P the associated change-of-basis matrix from the standard basis to the basis of
eigenvectors.
15.1 Review Problems
Webwork:
3 × 3 Example
Reading Problems 1 , 2 ,
Diagonalizing a symmetric matrix 3, 4
1. (On Reality of Eigenvalues)
(a) Suppose z = x + iy where x, y ∈ R, i = √ −1, and z = x − iy.
Compute zz and zz in terms of x and y. What kind of numbers
are zz and zz? (The complex number z is called the complex
conjugate of z).
(b) Suppose that λ = x + iy is a complex number with x, y ∈ R, and
that λ = λ. Does this determine the value of x or y? What kind
of number must λ be?
281