13.1. SYMMETRIC AND ORTHOGONAL MATRICES 291

13.1.2 Symmetric And Skew Symmetric Matrices

Definition 13.1.8 A real n× n matrix A, is symmetric if AT = A. If A = −AT , then A iscalled skew symmetric.

Theorem 13.1.9 The eigenvalues of a real symmetric matrix are real. The eigenvalues ofa real skew symmetric matrix are 0 or pure imaginary.

Proof: The proof of this theorem is in [13]. It is best understood as a special case ofmore general considerations. However, here is a proof in this special case.

Recall that for a complex number a+ ib, the complex conjugate, denoted by a+ ib isgiven by the formula a+ ib = a− ib. The notation, x will denote the vector which has everyentry replaced by its complex conjugate.

Suppose A is a real symmetric matrix and Ax = λx. Then

λxT x =(Ax)T x = xT AT x = xT Ax = λxT x.

Dividing by xT x on both sides yields λ = λ which says λ is real. (Why?)Next suppose A =−AT so A is skew symmetric and Ax = λx. Then

λxT x =(Ax)T x = xT AT x =−xT Ax =−λxT x

and so, dividing by xT x as before, λ =−λ . Letting λ = a+ ib, this means a− ib =−a− iband so a = 0. Thus λ is pure imaginary. ■

Example 13.1.10 Let A =

(0 −11 0

). This is a skew symmetric matrix. Find its eigen-

values.

Its eigenvalues are obtained by solving the equation det

(−λ −11 −λ

)= λ

2 +1 = 0.

You see the eigenvalues are ±i, pure imaginary.

Example 13.1.11 Let A =

(1 22 3

). This is a symmetric matrix. Find its eigenvalues.

Its eigenvalues are obtained by solving the equation, det

(1−λ 2

2 3−λ

)= −1−

4λ +λ2 = 0 and the solution is λ = 2+

√5 and λ = 2−

√5.

Definition 13.1.12 An n×n matrix A = (ai j) is called a diagonal matrix if ai j = 0 when-ever i ̸= j. For example, a diagonal matrix is of the form indicated below where ∗ denotesa number. 

∗ 0. . .

0 ∗

