A First Course in Linear Algebra, 2017a

7.4. Orthogonality 395
[ x(0)
y(0)
]
=
[ 2
2
]
Hint: form the matrix exponential e At and then the solution is e At C where C is the initial vector.
7.4 Orthogonality
7.4.1 Orthogonal Diagonalization
We begin this section by recalling some important definitions. Recall from Definition 4.122 that non-zero
vectors are called orthogonal if their dot product equals 0. A set is orthonormal if it is orthogonal and each
vector is a unit vector.
An orthogonal matrix U, from Definition 4.129, is one in which UU T = I. In other words, the transpose
of an orthogonal matrix is equal to its inverse. A key characteristic of orthogonal matrices, which will be
essential in this section, is that the columns of an orthogonal matrix form an orthonormal set.
We now recall another important definition.
Definition 7.48: Symmetric and Skew Symmetric Matrices
A real n × n matrix A, is symmetric if A T = A. If A = −A T , then A is called skew symmetric.
Before proving an essential theorem, we first examine the following lemma which will be used below.
Lemma 7.49: The Dot Product
Let A = [ ]
a ij be a real symmetric n × n matrix, and let⃗x,⃗y ∈ R n .Then
A⃗x •⃗y =⃗x • A⃗y
Proof. This result follows from the definition of the dot product together with properties of matrix multiplication,
as follows:
A⃗x •⃗y = ∑a kl x l y k
k,l
= ∑(a lk ) T x l y k
k,l
= ⃗x • A T ⃗y
= ⃗x • A⃗y
The last step follows from A T = A, sinceA is symmetric.
♠
We can now prove that the eigenvalues of a real symmetric matrix are real numbers. Consider the
following important theorem.