[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）19 (1002ﾚｽ)
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739(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 04/04(金)10:26 ID:nFnX0O4C(1/4) AAS
>>734 補足
>4 Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402.
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P62
Theorem 1.3.12. Let A, B ∈ Mn be diagonalizable. Then A and B commute if and only if they are simultaneously diagonalizable.
Proof.
Assume that A and B commute, perform a similarity transformation on both A and B that diagonalizes A (but not necessarily B) and groups together any repeated eigenvalues of A. Ifμ1,...,μd are the distinct eigenvalues of A and n1,...,nd are their respective multiplicities, then we may assume that
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740: 04/04(金)10:26 ID:nFnX0O4C(2/4) AAS
つづき
Proof. If F is simultaneously diagonalizable, then it is a commuting family by a previous exercise. We prove the converse by induction on n.Ifn = 1, there is nothing to prove since every family is both commuting and diagonal. Let us suppose that n ≥ 2 and that, for each k = 1,2,...,n − 1,anycommutingfamilyofk-by-k diagonalizable matrices is simultaneously diagonalizable. If every matrix in F is a scalar matrix, there is nothing to prove, so we may assume that A ∈ F is a given n-by-n diagonalizable matrix with distinct eigenvalues λ1,λ2,...,λk and k ≥ 2, that AB = BAfor every B ∈F, and that each B ∈ F is diagonalizable. Using the argument in (1.3.12), we reduce to the case in which A has the form (1.3.13). Since every B ∈ F commutes with A, (0.7.7) ensures that each B ∈ F has the form (1.3.14). Let B, ˆ B ∈ F, so B = B1⊕···⊕Bk and ˆ B = ˆ B1 ⊕···⊕ˆ Bk, in which each of Bi, ˆ Bi has the same size and that size is at most n − 1. Commutativity and diagonalizability of B and ˆ B imply commutativity and diagonalizability of Bi and ˆ Bi for each i = 1,...,d. By the induction hypothesis, there are k similarity matrices T1, T2,...,Tk of appropriate size,
each of which diagonalizes the corresponding block of every matrix in F. Then the direct sum (1.3.15) diagonalizes every matrix in F. Wehaveshownthat there is a nonsingular T ∈ Mn such that T−1BT is diagonal for every B ∈ F. Then T−1A0T = Pdiag(λ1,...,λn)PT for some permutation matrix P, PT(T−1A0T)P = (TP)−1A0(TP) = diag(λ1,...,λn) and (TP)−1B(TP) = PT(T−1BT)P is diagonal for every B ∈ F (0.9.5). □
Remarks: We defer two important issues until Chapter 3: (1) Given A, B ∈ Mn, how can we determine if A is similar to B? (2) How can we tell if a given matrix is diagonalizable without knowing its eigenvectors?
Although AB and BA need not be the same (and need not be the same size even when both products are defined), their eigenvalues are as much the same as possible. Indeed, if A and B arebothsquare,then ABand BAhaveexactlythesameeigenvalues. These important facts follow from a simple but very useful observation.
(引用終り)
以上
741(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 04/04(金)10:54 ID:nFnX0O4C(3/4) AAS
>>739 ついでに補足
”対角化可能であるための必要十分条件”
結論：やっぱ 固有値は大事だ！ｗ ；ｐ）
外部ﾘﾝｸ:ja.wikipedia.org
対角化（たいかくか、diagonalization[1]）とは、正方行列を適当な線形変換によりもとの行列と相似な対角行列に変形することを言う。
対角化により変換において本質的には無駄な計算を省くことで計算量を大幅に減らすことができる。
対角化可能であるための必要十分条件
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744: 現代数学の系譜 雑談 ◆yH25M02vWFhP 04/04(金)15:57 ID:nFnX0O4C(4/4) AAS
下記 那須 弘和先生分かり易い
学歴 平成 9年 3月 名古屋大学 理学部数学科卒業か
外部ﾘﾝｸ:fuji.ss.u-tokai.ac.jp
那須 弘和 Hirokazu Nasu
東海大学 理学部 情報数理学科 学歴 平成 9年 3月 名古屋大学 理学部数学科卒業
外部ﾘﾝｸ[html]:fuji.ss.u-tokai.ac.jp
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