純粋・応用数学・数学隣接分野（含むガロア理論）19

[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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734: １３２人目の素数さん [] 2025/04/04(金) 07:58:00.13 ID:CsC7EptL

つづき

https://en.wikipedia.org/wiki/Commuting_matrices
Commuting matrices

Characterizations and properties
・Two diagonalizable matrices
A and B commute ( AB=BA if they are simultaneously diagonalizable (that is, there exists an invertible matrix
P such that both  P^{-1}AP and  P^{-1}BP are diagonal).[4]: p. 64 
The converse is also true; that is, if two diagonalizable matrices commute, they are simultaneously diagonalizable.[5]
But if you take any two matrices that commute (and do not assume they are two diagonalizable matrices) they are simultaneously diagonalizable already if one of the matrices has no multiple eigenvalues.[6]

References
4  Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402.
5  Without loss of generality, one may suppose that the first matrix
A=(a_{i,j}) is diagonal. In this case, commutativity implies that if an entry
b_{i,j of the second matrix is nonzero, then  a_{i,i}=a_{j,j}.
After a permutation of rows and columns, the two matrices become simultaneously block diagonal.
In each block, the first matrix is the product of an identity matrix, and the second one is a diagonalizable matrix. So, diagonalizing the blocks of the second matrix does change the first matrix, and allows a simultaneous diagonalization.
6  "Proofs Homework Set 10 MATH 217 — WINTER 2011" (PDF). Retrieved 10 July 2022. http://www.math.lsa.umich.edu/~tfylam/Math217/proofs10-sol.pdf （Thomas Lam I am Professor of Mathematics at the University of Michigan. https://dept.math.lsa.umich.edu/~tfylam/ Math 217 Winter 2011 - Linear Algebra）
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1725190538/734

739: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/04/04(金) 10:26:20.63 ID:nFnX0O4C

>>734 補足
>4 Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402.

これの海賊版PDFが見つかった
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
略す

P63
Observation 1.3.18. Suppose that n ≥ 2. A given A ∈ Mn is similar to a block triangular matrix of the form (1.3.17) if and only if some nontrivial subspace of Cn is A-invariant. Moreover, if W ⊆ Cn isanonzero A-invariantsubspace,thensomevector in W is an eigenvector of A. A given family F ⊆ Mn is reducible if and only if there is some k ∈{2,...,n −1} and a nonsingular S ∈ Mn such that S−1AS has the form (1.3.17) for every A ∈ F.

The following lemma is at the heart of many subsequent results.

Lemma1.3.19. Let F ⊂ Mn beacommutingfamily. Then some nonzero vector in Cn is an eigenvector of every A ∈ F.
Proof. 略す

P64
Lemma 1.3.19 concerns commuting families of arbitrary nonzero cardinality. Our next result shows that Theorem 1.3.12 can be extended to arbitrary commuting families of diagonalizable matrices.

Definition 1.3.20. A family F ⊂ Mn is said to be simultaneously diagonalizable if there is a single nonsingular S ∈ Mn such that S−1AS is diagonal for every A ∈ F.

Theorem 1.3.21. Let F ⊂ Mn be a family of diagonalizable matrices. Then F is a commuting family if and only if it is a simultaneously diagonalizable family. Moreover, for any given A0 ∈ F and for any given ordering λ1,...,λn of the eigenvalues of A0, there is a nonsingular S ∈ Mn such that S−1A0S = diag(λ1,...,λn) and S−1BS is diagonal for every B ∈ F.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1725190538/739

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