A First Course in Linear Algebra, 2017a

7.4. Orthogonality 397
Solution. First notice that A is skew symmetric. By Theorem 7.51, the eigenvalues will either equal 0 or
be pure imaginary. The eigenvalues of A are obtained by solving the usual equation
[ ]
x 1
det(xI − A)=det = x 2 + 1 = 0
−1 x
Hence the eigenvalues are ±i, pure imaginary.
♠
Consider the following example.
Example 7.53: Eigenvalues of a Symmetric Matrix
[ ] 1 2
Let A = . Find its eigenvalues.
2 3
Solution. First, notice that A is symmetric. By Theorem 7.50, the eigenvalues will all be real.
eigenvalues of A are obtained by solving the usual equation
[ ]
x − 1 −2
det(xI − A)=det
= x 2 − 4x − 1 = 0
−2 x − 3
The eigenvalues are given by λ 1 = 2 + √ 5andλ 2 = 2 − √ 5 which are both real.
The
♠
Recall that a diagonal matrix D = [ ]
d ij is one in which dij = 0 whenever i ≠ j. In other words, all
numbers not on the main diagonal are equal to zero.
Consider the following important theorem.
Theorem 7.54: Orthogonal Diagonalization
Let A be a real symmetric matrix. Then there exists an orthogonal matrix U such that
U T AU = D
where D is a diagonal matrix. Moreover, the diagonal entries of D are the eigenvalues of A.
We can use this theorem to diagonalize a symmetric matrix, using orthogonal matrices. Consider the
following corollary.
Corollary 7.55: Orthonormal Set of Eigenvectors
If A is a real n × n symmetric matrix, then there exists an orthonormal set of eigenvectors,
{⃗u 1 ,···,⃗u n }.
Proof. Since A is symmetric, then by Theorem 7.54, there exists an orthogonal matrix U such that U T AU =
D, a diagonal matrix whose diagonal entries are the eigenvalues of A. Therefore, since A is symmetric and
all the matrices are real,
D = D T = U T A T U = U T A T U = U T AU = D