Page 34 - Vector Analysis

30 CHAPTER 1. Linear Algebra

Let B be the standard basis of Fn, and tukuk8=1 be a sequence of vectors in Fn such that
}uk}Fn = 1 , and lim (Luk, uk)Fn = M . Then (1.5) implies }M uk ´ Luk}Fn Ñ 0 as k Ñ 8 .

kÑ8

Since M R σ(L) , M In ´ [L]B is invertible; thus

[uk]B = (M In ´ [L]B)´1(M [uk]B ´ [L]B[uk]B) Ñ 0 in Fn

which contradicts to }uk}Fn = 1 for all k P N. Hence M P σ(L). Similarly, m P σ(L). ˝

Definition 1.96 (Diagonalizable linear maps). Let V be a finite dimensional vector spaces
over a scalar field F. A linear map L : V Ñ V is said to be diagonalizable if there is a
basis B of V such that each v P B is an eigenvector of L.

Theorem 1.97. Let L P B(Rn) be symmetric. Then there exists an orthonormal basis of
Rn consisting of eigenvectors of L.

Example 1.98 (The 2-norm of matrices). Let (¨, ¨)Rk denote the inner product in Euclidean
space Rk, and A P M(m, n; R). Since ATA is a symmetric n ˆ n matrix, it is diagonalizable
by an orthonormal matrix P ; that is, ATA = P ΛP T for some orthonormal n ˆ n matrix P
and diagonal n ˆ n matrix Λ = [λiδij]. Therefore,

}Ax}22 = (Ax, Ax)Rm = (x, ATAx)Rn = (x, P ΛP Tx)Rn = (P Tx, ΛP Tx)Rn

which implies that

sup }Ax}22 = sup (P Tx, ΛP Tx)Rn = sup (y, Λy)Rn (Let y = P Tx, then }y}2 = 1)

}x}2=1 }x}2=1 }y}2=1

= sup (λ1y12 + λ2y22 + ¨ ¨ ¨ + λnyn2)

}y}2=1

= max ␣λ1, ¨ ¨ ¨ , λn( = maximum eigenvalue of ATA .

As a consequence, }A}2 = amaximum eigenvalue of ATA.

1.9 The Einstein Summation Convention

In mathematics, especially in applications of linear algebra to physics, the Einstein sum-
mation convention is a notational convention that implies summation over a set of indexed
terms in a formula, thus achieving notational brevity. According to this convention, when
an index variable appears twice in a single term it implies summation of that term over all