A First Course in Linear Algebra, 2017a

396 Spectral Theory
Theorem 7.50: Orthogonal Eigenvectors
Let A be a real symmetric matrix. Then the eigenvalues of A are real numbers and eigenvectors
corresponding to distinct eigenvalues are orthogonal.
Proof. Recall that for a complex number a + ib, the complex conjugate, denoted by a + ib is given by
a + ib = a − ib. The notation, ⃗x will denote the vector which has every entry replaced by its complex
conjugate.
Suppose A is a real symmetric matrix and A⃗x = λ⃗x. Then
λ⃗x T ⃗x = ( A⃗x ) T
⃗x =⃗x
T
A T ⃗x =⃗x T A⃗x = λ⃗x T ⃗x
Dividing by ⃗x T ⃗x on both sides yields λ = λ which says λ is real. To do this, we need to ensure that
⃗x T ⃗x ≠ 0. Notice that⃗x T ⃗x = 0 if and only if⃗x =⃗0. Since we chose⃗x such that A⃗x = λ⃗x,⃗x is an eigenvector
and therefore must be nonzero.
Now suppose A is real symmetric and A⃗x = λ⃗x, A⃗y = μ⃗y where μ ≠ λ. ThensinceA is symmetric, it
follows from Lemma 7.49 about the dot product that
λ⃗x •⃗y = A⃗x •⃗y =⃗x • A⃗y =⃗x • μ⃗y = μ⃗x •⃗y
Hence (λ − μ)⃗x•⃗y = 0. It follows that, since λ −μ ≠ 0, it must be that⃗x•⃗y = 0. Therefore the eigenvectors
form an orthogonal set.
♠
The following theorem is proved in a similar manner.
Theorem 7.51: Eigenvalues of Skew Symmetric Matrix
The eigenvalues of a real skew symmetric matrix are either equal to 0 or are pure imaginary numbers.
Proof. First, note that if A = 0 is the zero matrix, then A is skew symmetric and has eigenvalues equal to
0.
Suppose A = −A T so A is skew symmetric and A⃗x = λ⃗x. Then
λ⃗x T ⃗x = ( A⃗x ) T
⃗x =⃗x
T
A T ⃗x = −⃗x T A⃗x = −λ⃗x T ⃗x
and so, dividing by⃗x T ⃗x as before, λ = −λ. Letting λ = a + ib, this means a − ib = −a − ib and so a = 0.
Thus λ is pure imaginary.
♠
Consider the following example.
Example 7.52: Eigenvalues of a Skew Symmetric Matrix
[ ]
0 −1
Let A = . Find its eigenvalues.
1 0