334 CHAPTER 13. MATRICES AND THE INNER PRODUCT

and if you write the quadratic form in terms of the primed variables, there will be no mixedterms. Any symmetric real matrix is Hermitian and is therefore normal. From Corollary13.1.6, it follows there exists a real unitary matrix U, (an orthogonal matrix) such thatUT AU = D a diagonal matrix. Thus in the quadratic form, 13.3

(x y z

)A

 xyz

 =(

x′ y′ z′)

UT AU

 x′

y′

z′

=

(x′ y′ z′

)D

 x′

y′

z′

and in terms of these new variables, the quadratic form becomes

λ 1(x′)2

+λ 2(y′)2

+λ 3(z′)2

where D= diag(λ 1,λ 2,λ 3) . Similar considerations apply equally well in any other dimen-sion. For the given example, −

12

√2 0 1

2

√2

16

√6 1

3

√6 1

6

√6

13

√3 − 1

3

√3 1

3

√3

 3 −4 1−4 0 −41 −4 3

 ·− 1√

21√6

1√3

0 2√6− 1√

31√2

1√6

1√3

=

 2 0 00 −4 00 0 8

and so if the new variables are given by

− 1√2

1√6

1√3

0 2√6− 1√

31√2

1√6

1√3

 x′

y′

z′

=

 xyz

 ,

it follows that in terms of the new variables the quadratic form is 2(x′)2−4(y′)2 +8(z′)2 .You can work other examples the same way.

13.3 The Estimation Of EigenvaluesThere are ways to estimate the eigenvalues for matrices. The most famous is known asGerschgorin’s theorem. This theorem gives a rough idea where the eigenvalues are justfrom looking at the matrix.

Theorem 13.3.1 Let A be an n×n matrix. Consider the n Gerschgorin discs defined as

Di ≡

{λ ∈ C : |λ −aii| ≤∑

j ̸=i

∣∣ai j∣∣} .

Then every eigenvalue is contained in some Gerschgorin disc.