Some Asymptotic Behaviors Associated with Matrix Decompositions

Mathematics & Statistics Auburn University, Alabama, USA

Beurling- ... QR May, 2008 Cholesky QR iteration Some Asymptotic Behaviors Associated Aluthge

with Matrix Decompositions Home Page

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JJ II Speaker: Tin-Yau Tam (譚譚譚天天天祐祐祐) J I

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Go Back Colloquium Full Screen

[email protected] Quit Beurling- ... QR Discuss some Matrix Decompositions Cholesky QR iteration and Aluthge their Asymptotic Behaviors Home Page

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J I joint work with Huajun Huang (黄华君) Page 2 of 24

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Quit Outline Beurling- ... QR Cholesky QR iteration SVD Aluthge

QR decomposition Home Page

Title Page Cholesky decomposition JJ II J I QR iteration Page 3 of 24 Aluthge iteration Go Back Full Screen

Quit 1. Beurling-Gelfand Spectral radius formula

Theorem 1.1. (Gelfand, 1941) Let A ∈ Cn×n.

m 1/m lim kA k = r(A) (1) Beurling- ... m→∞ QR Cholesky where r(A) is the spectral radius of A and kAk is the spectral norm of A. QR iteration Aluthge • obtained earlier by Beurling (1938).

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Spectral radius: r(A) is the maximum eigenvalue modulus. Title Page 2-norm: kvk = (v∗v)1/2 2 JJ II

Spectral norm: kAk := maxkvk2=1 kAvk2 J I

Page 4 of 24 m 1/m Remark: limm→∞ kA k depends obviously on metric properties of A. Go Back

However the spectrum and the thus spectral radius of an A are deﬁned in Full Screen terms of the algebraically, regardless of any metric (or topological) consid- Close eration. A very remarkable feature of the spectral radius formula. Quit A proof:

Choose λ ∈ σ(A) such that r(A) = |λ| and v a corresponding eigenvector. Beurling- ... QR Then Cholesky QR iteration Aluthge kAkkvk ≥ kAvk = kλ vk = r(A)kvk.

One has Home Page

kAk ≥ r(A). Title Page Now JJ II kAkm ≥ kAmk ≥ r(Am) = rm(A). J I Page 5 of 24

So Go Back m 1/m kAk ≥ kA k ≥ r(A), m = 1, 2,... Full Screen

Quit m 1/m Let α be any limit point of {kA k }m∈N, that is,

kAmik1/mi → α, as i → ∞.

Clearly r(A) ≤ α. If r(A) < α then there would exist α0 such that

0 Beurling- ... r(A) < α < α. QR Cholesky Clearly QR iteration Aluthge r(A/α0) = r(A)/α0 < 1 and thus Home Page 0 mi (A/α ) → 0, as i → ∞ Title Page

For ε > 0 there is N(ε) ∈ N such that JJ II J I Ami k k < ε Page 6 of 24 α0 Go Back for every i > N(ε). So for a ﬁxed ε, Full Screen

m Close α kAmik1/mi A i 1 < = lim = lim k k1/mi ≤ lim ε1/mi = 1, α0 i→∞ α0 i→∞ α0 i→∞ Quit a contradiction. Banach algebra A Banach algebra A is a complex algebra and a Banach space with respect Beurling- ... QR to a norm that satisﬁes the submultiplicative property: Cholesky QR iteration Aluthge kxyk ≤ kxk kyk, x, y ∈ A and A contains a unit element e such that Home Page

Title Page kek = 1 JJ II and J I xe = ex = x Page 7 of 24 Go Back for all x ∈ A. Full Screen

Quit Example: Cn×n is a Banach algebra with In as the identity element with respect to the spectral norm. Beurling- ... QR Cholesky Example: H = a Banach space. Then B(H) = the algebra of all bounded QR iteration linear operators on H is a Banach algebra with respect to the usual operator Aluthge norm. The identity operator I is the unit element. Every closed subalgebra Home Page of B(H) that contains I is also a Banach algebra. Title Page

JJ II Spectrum σ(x) of x ∈ A is the set of all λ ∈ C such that λe − x is not J I invertible. Page 8 of 24 Spectral radius of x: Go Back

r(x) = sup{|λ| : λ ∈ σ(x)} Full Screen

Quit Beurling- ... QR Theorem 1.2. Let A be a Banach algebra and x ∈ A. Cholesky QR iteration 1. The spectrum σ(x) is compact and nonempty. Aluthge

2. Home Page m 1/m m 1/m lim kx k = inf kx k = r(x) Title Page m→∞ m≥1 Remark: r(x) ≤ kxk is contained in the formula. JJ II J I

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Quit Singular values of A ∈ Cn×n are the square roots of eigenvalues of the positive semi-deﬁnite matrix A∗A.

Fact: kAk = s1(A), the largest singular value of A. Beurling- ... Arrange: QR Cholesky QR iteration s1(A) ≥ s2(A) ≥ · · · ≥ sn(A), |λ1(A)| ≥ |λ2(A)| ≥ · · · ≥ |λn(A)|. Aluthge

Home Page SVD: Given A ∈ Cn×n, there exist unitary U, V such that Title Page

A = Udiag (s1, . . . , sn)V. JJ II

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Go Back Rewrite Beurling-Gelfand : Full Screen

m 1/m m 1/m Close lim kA k = r(A) ⇔ lim [s1(A )] = |λ1(A)| m→∞ 2 m→∞ Quit Yamamoto (1967):

m 1/m lim [si(A )] = |λi(A)|, i = 1, . . . , n. m→∞ Beurling- ... • QR a natural generalization of Beurling-Gelfand (ﬁnite dim. case). Cholesky • Loesener (1976) rediscovered Yamamoto QR iteration Aluthge • Mathias (1990 another proof) • Johnson and Nylen (1990 generalized singular values) Home Page • Nylen and Rodman (1990 Banach algebra) Title Page

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Extension to semisimple Lie group: Page 11 of 24

T.Y. Tam and H. Huang, An extension of Yamamoto’s theorem on the eigen- Go Back values and singular values of a matrix, Journal of Math. Soc. Japan, 58 Full Screen

(2006) 1197-1202. Close

Quit 2. QR decomposition

Given nonsingular A ∈ Cn×n, recall QR decomposition Beurling- ... QR A = QR Cholesky QR iteration Set Aluthge

a(A) := diag (r11, . . . , rnn) Home Page A where is written in column form Title Page

JJ II A = [A1| · · · |An] J I

Geometric interpretation of a(A): Page 12 of 24 rii is the distance between Ai and the span of A1,...,Ai−1, i = 2, . . . , n. Go Back

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Remark: QR decomposition is a matrix version of Gram-Schmidt orthonor- Close malization process. Quit Theorem 2.1. (Huang and Tam, 2007 LAA) Given A, X, B ∈ Cn×n nonsingular. Let X = Y −1JY be the Jordan decomposition of X, where J is the Jordan form of X, Beurling- ... QR diag J = diag (λ1, . . . , λn) Cholesky QR iteration Aluthge satisfying |λ1| ≥ · · · ≥ |λn|. Then

m 1/m lim a(AX B) = diag (|λω(1)|,..., |λω(n)|), m→∞ Home Page where the permutation ω is uniquely determined by the Gelfand-Naimark Title Page decomposition of YB = LωU: JJ II J I

rank ω(i|j) = rank (YB)(i|j), 1 ≤ i, j ≤ n. Page 13 of 24

Go Back Here ω(i|j) denotes the submatrix formed by the ﬁrst i rows and the ﬁrst j Full Screen columns of ω, 1 ≤ i, j ≤ n. Close

Quit Corollary 2.1. Let k · k be a norm on n and i = 1, . . . , n. The m-th root of Beurling- ... C QR m the distance in k · k between the i-th column of AX B and the span of the Cholesky m QR iteration previous i − 1 columns of AX B converges to |λω(i)|. Aluthge

Home Page Extension to semisimple Lie group: Title Page H. Huang and T.Y. Tam, An asymptotic behavior of QR decomposition, Lin- JJ II ear Algebra and Its Applications, 424 (2007) 96-107. J I H. Huang and T.Y. Tam, An asymptotic result on the a-component in Iwa- Page 14 of 24 sawa decomposition, Journal of Lie Theory, 17 (2007) 469-479. Go Back

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Quit Another way to get ω:

−1 Theorem 2.2. Suppose X = Y TY ∈ GLn(C) where T ∈ Cn×n is a lower Beurling- ... triangular matrix and the diagonal entries of T satisfy that |t | ≥ |t | ≥ QR 11 22 Cholesky · · · ≥ |tnn|, so that λk := tkk are the eigenvalues of X with |λ1| ≥ |λ2| ≥ QR iteration Aluthge · · · ≥ |λn|. Then

m 1/m Home Page lim [a(AX B)] = diag (|λω(1)|,..., |λω(n)|), m→∞ Title Page where ω is a permutation determined by the Gelfand-Naimark decomposition JJ II YB = LωU of YB, such that L is unit lower triangular and U is upper J I triangular. Page 15 of 24

m 1/m Go Back Idea of proof: Establish limm→∞[a1(AX B)] = |λω(1)|. Then use funda- mental representations. Full Screen Close

Quit Numerical experiments: Computing the discrepancy between

m 1/m Beurling- ... [a(A )] and |λ(A)| QR Cholesky QR iteration of randomly generated A ∈ GLn( ). C Aluthge

The graph of Home Page

100 m 1/m k[a(A )] − diag (|λ1|,..., |λn|)k2 Title Page

JJ II 80 versus m (m = 1,..., 100) J I 60 Page 16 of 24

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Quit If we consider m 1/m |a1(A ) − |λ1(A)|| Beurling- ... for the above example, convergence occurs. QR Cholesky QR iteration Aluthge

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Quit 3. Cholesky decomposition

Cholesky decomposition: Given a positive deﬁnite A ∈ Cn×n,

A = R∗R

Beurling- ... where R is an upper triangular matrix with positive diagonal entries. QR Cholesky Spectral theorem: There is a unitary matrix Y such that QR iteration Aluthge Y AY −1 = Λ

Home Page is diagonal with nonincreasing diagonal entries. Title Page

Theorem 3.1. (Huang and Tam, 2007) Let A ∈ Cn×n be positive deﬁnite. JJ II −1 Let A = Y ΛY where Y ∈ U(n) and λ1 ≥ · · · ≥ λn > 0. Then J I

Page 18 of 24 m 1/m lim d(A ) = diag (λω(1), . . . , λω(n)) i→∞ Go Back

Full Screen where ω ∈ Sn is uniquely determined by the Gelfand-Naimark decomposition Y = LωU of Y . Close Quit H. Huang and T.Y. Tam, International J. of Information & Systems Sciences on Matrix Analysis and Applications, 4 (2008), 148-159. 4. QR iteration Beurling- ... QR Cholesky Deﬁne a sequence {Am}m∈N of matrices with QR iteration Aluthge A1 := A and Home Page

Aj+1 := RjQj if Aj = QjRj, j = 1, 2,... Title Page

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J I A1 = Q1R1,A2 = R1Q1 = Q2R2,A3 = R2Q2,... Page 19 of 24

Notice that Go Back −1 Am+1 = Qm AmQm. Full Screen Similarity → the eigenvalues of A are ﬁxed in the process Close Quit Theorem 4.1. (Francis, 1961/62, Huang and Tam, 2005 LAA) Suppose that the moduli of the eigenvalues λ1, . . . , λn of A ∈ GLn(C) are distinct:

|λ1| > |λ2| > ··· > |λn| (> 0). Beurling- ... QR Let Cholesky A = Y −1 (λ , . . . , λ )Y. QR iteration diag 1 n Aluthge Assume Y = LωU, Home Page

Title Page where ω is a permutation, L is lower ∆ and U is unit upper ∆. Then JJ II

1. the strictly lower triangular part of Am converges to zero. J I

Page 20 of 24 2. diag Am → diag (λω(1), . . . , λω(n)). Go Back

• Kublanovskaja (1961) also obtained the result for Y = LU. Full Screen

Huajun Huang and Tin-Yau Tam, On the QR iterations of real matrices, Lin- Close ear Algebra and Its Applications, 408 (2005) 161-176. Quit 5. λ-Aluthge iteration Beurling- ... QR Given 0 < λ < 1, the λ-Aluthge transform of A ∈ Cn×n: Cholesky QR iteration λ 1−λ Aluthge ∆λ(A) := P UP ,

where A = UP is the polar decomposition of A, that is, U is unitary and P Home Page is positive semideﬁnite. Title Page

Theorem 5.1. (Antezana, Massey and Stojanoff, 2005) JJ II J I Let A ∈ Cn×n and 0 < λ < 1. Page 21 of 24 m 1. Any limit point of the λ-Aluthge sequence {∆λ (A)}m∈N is normal, with Go Back

eigenvalues λ1(A), . . . , λn(A). Full Screen

m Close 2. limm→∞ k∆λ (A)k = r(A). Quit m • Yamazaki (2003) obtained limm→∞ k∆λ (T )k = r(T ) for Hilbert space bounded operator T when λ = 1/2. Beurling- ... QR Cholesky Theorem 5.2. (Tam, 2007) Let H be a Hilbert space. Suppose B(H) is a QR iteration Banach algebra with unit I with respect to the unitarily invariant norm ||| · ||| . Aluthge Let T ∈ B(H) be invertible and 0 < λ < 1. Then Home Page n lim ||| ∆λ(T ) ||| = r(T ). Title Page n→∞ JJ II Conjecture: Given A ∈ Cn×n, 0 < λ < 1, the λ-Aluthge sequence m J I {∆λ (A)}m∈ converges. N Page 22 of 24 Tin-Yau Tam, λ-Aluthge iteration and spectral radius, to appear in Integral Go Back Equations and Operator Theory. Full Screen

Quit Theorem 5.3. (Huang and Tam, 2007) Let A ∈ Cn×n and 0 < λ < 1. If the nonzero eigenvalues of A have distinct moduli, then the λ-Aluthge se- m Beurling- ... quence {∆λ (A)}m∈N converges to a normal matrix with the same eigenval- QR ues (counting multiplicity) as A. Cholesky QR iteration Huajun Huang and Tin-Yau Tam, On the convergence of Aluthge sequence, Aluthge Operators and Matrices, 1 (2007) 121-142. Home Page So the conjecture is true for a dense subset of Cn×n: the matrices of distinct Title Page eigenvalue moduli. JJ II

Theorem 5.4. (Antezana, Pujuls and Stojanoff, 2007) If A ∈ Cn×n is diag- J I

m Page 23 of 24 onalizable, then the Aluthge sequence {∆1/2(A)}m∈N converges to a normal matrix with the same eigenvalues (counting multiplicity) as A. Go Back

Full Screen The proof of APS’s result uses differential geometry. Close

Quit T HAN K YOU FOR YOURATTENTION

Beurling- ... QR Cholesky QR iteration Aluthge

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