304 CHAPTER 13. MATRICES AND THE INNER PRODUCT

Proof: Use block multiplication to write(AB 0B 0

)(I A0 I

)=

(AB ABAB BA

)(

I A0 I

)(0 0B BA

)=

(AB ABAB BA

).

Therefore, (I A0 I

)−1(AB 0B 0

)(I A0 I

)=

(0 0B BA

)Since the two matrices above are similar it follows that(

0 0B BA

)and (

AB 0B 0

)have the same characteristic polynomials. Therefore, noting that BA is an n×n matrix andAB is an m×m matrix,

tm det(tI−BA) = tn det(tI−AB)

and so det(tI−BA) = pBA (t) = tn−m det(tI−AB) = tn−m pAB (t). ■

13.2.2 Orthonormal Bases, Gram Schmidt ProcessNot all bases for Fn are created equal. Recall F equals either C or R and the dot product isgiven by

x ·y≡ (x,y)≡ ⟨x,y⟩= ∑j

x jy j.

The best bases are orthonormal. Much of what follows will be for Fn in the interest ofgenerality.

Definition 13.2.5 Suppose {v1, · · · ,vk} is a set of vectors in Fn. It is an orthonormal set if

vi ·v j = δ i j =

{1 if i = j0 if i ̸= j

Every orthonormal set of vectors is automatically linearly independent.

Proposition 13.2.6 Suppose {v1, · · · ,vk} is an orthonormal set of vectors. Then it is lin-early independent.

Proof: Suppose ∑ki=1 civi = 0. Then taking dot products with v j,

0 = 0 ·v j = ∑i

civi ·v j = ∑i

ciδ i j = c j.

Since j is arbitrary, this shows the set is linearly independent as claimed. ■It turns out that if X is any subspace of Fm, then there exists an orthonormal basis for

X . This follows from the use of the next lemma applied to a basis for X .