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Matrix Canonical Forms Matrix Canonical Forms Roger Horn University of Utah ICTP School: Linear Algebra: Friday, June 26, 2009 ICTP School: Linear Algebra: Friday, June 26, 2009 1 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Canonical forms for congruence and *congruence Congruence A SAST (change variables in quadratic form xT Ax) ! *Congruence A SAS (change variables in Hermitian form x Ax) ! Congruence and *congruence are simpler than similarity: no inverses; identical row and column operations for congruence (complex conjugates for *congruence). The singular and nonsingular canonical structures are fundamentally di¤erent. ICTP School: Linear Algebra: Friday, June 26, 2009 2 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Example: *Congruence for Hermitian matrices Sylvester’sInertia Theorem (1852): Two Hermitian matrices are *congruent if and only if they have the same number of positive eigenvalues and the same number of negative eigenvalues (and hence also the same number of zero eigenvalues). Reformulate: Two Hermitian matrices are *congruent if and only if they have the same number of eigenvalues on each of the two open rays tei0 : t > 0 and teiπ : t > 0 in the complex plane. f g i0 f iπ g Canonical form: e In+ e In 0n0 ICTP School: Linear Algebra: Friday, June 26, 2009 3 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Example: *Congruence for normal matrices Unitary *congruence: Two normal matrices are unitarily *congruent (unitarily similar!) if and only if they have the same eigenvalues. Ikramov (2001): Two normal matrices are *congruent if and only if they have the same number of eigenvalues on each open ray teiθ : t > 0 , θ [0, 2π) in the complex plane. f g 2iθ iθ Canonical form: (e 1 In ) (e k In ) 0n , θ1 θk 0 0 θ1 < < θk < 2π What comes next? Find a theorem about *congruence of general n- by-n matrices that includes Sylvester’sand Ikramov’stheorems as special cases. Find an analogous theorem for congruence. ICTP School: Linear Algebra: Friday, June 26, 2009 4 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Example: Congruence of complex symmetric matrices Two complex symmetric matrices are congruent if and only if they have the same rank. Why? We know that if A = AT then there is a unitary U such that T A = UΣU . If rank A = r, let D = diag(pσ1, ... , pσr , 1, ... , 1). T Then A = (UD)(Ir 0n r )(UD) Canonical form: Ir 0n r What comes next? Find a theorem about congruence of general n- by-n matrices that includes this observation as a special case. ICTP School: Linear Algebra: Friday, June 26, 2009 5 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Nullities are *congruence and congruence invariants rank A = rank SAS dim N(A) = dim N(SAS ) = dim N(SAS ) = dim N(A) Moreover, dim(N(A) N(A)) = dim(N(SAS ) N(SAS )) \ \ rank A = rank SAST dim N(A) = dim N(SAST ) = dim N(SAT ST ) = dim N(AT ) Moreover, T T dim(N(A) N(A )) = dim(N(SAS ) N(SA S )) \ \ These observations are the key to the regularization algorithm: Reduce A by congruence (respectively, *congruence) to the direct sum of a nonsingular part and a singular part. Deal with each part separately. ICTP School: Linear Algebra: Friday, June 26, 2009 6 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Regularization for *congruence and congruence Each singular A is *congruent to in which A S is nonsingular A = Jn (0) Jn (0) S 1 p block sizes ni uniquely determined by the *congruence class of A. *congruence class of uniquely determined by the *congruence class of A A Same for congruence. ICTP School: Linear Algebra: Friday, June 26, 2009 7 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Regularization algorithm: Step 1 Step 1: Choose nonsingular S so the top rows of SA are independent and the bottom m1 rows are zero, then form SAS ; partition it so that the upper left block is square: A independent rows A SA = 0 7! 0 A0S M N (M is square) SAS = = 7! 0 0 0m 1 m1 = dim N(A) = dim N(A) = the nullity of A m2 = rank N m1 m2 = nullity of N = dim(N(A) N(A)) = the normal nullity of A \ m2 = the non-normal nullity of A = the nullity of M Same for congruence. ICTP School: Linear Algebra: Friday, June 26, 2009 8 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Regularization algorithm: Step 2 Step 2: Choose nonsingular R so the top rows of RN are zero and the bottom m2 rows are independent: 0 RN = E m2 independent rows M N M N (R I ) (R I ) 0 0m 7! 0 0m 1 1 A(1) B 0 RMR RN = = CD E m2 0 0m1 2 3g 0 0 m1 m1 g 4 5 D is m2-by-m2 and A(1) is strictly smaller than A. Same for congruence. ICTP School: Linear Algebra: Friday, June 26, 2009 9 Roger Horn (University of Utah) Matrix Canonical Forms / 22 The regularizing decomposition If A(1) is nonsingular or missing, stop. If A(1) is present and singular, repeat steps 1 and 2 on A(1) to obtain m3 (the nullity of A(1)) and m4 (the non-normal nullity of A(1)). Repeat (for τ steps, say) until either a nonsingular block (m2τ+1 = 0) or an empty block is obtained. A singular A Mn is congruent to in which the regular part is nonsingular2 and A S A [m1 m2 ] [m2 m3 ] [m2τ 1 m2τ ] [m2τ ] = J1 J2 J2τ 1 J2τ S [p] J := Jk Jk (p direct summands) k The integers m1 m2 m2τ 0, as well as the congruence class of , are uniquely determined by the congruence class of A. A ICTP School: Linear Algebra: Friday, June 26, 2009 10 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Example 1 0 1 1 MN 0 1 1 A = 0 0 1 = ; M = , N = 2 3 [0 0] 0 0 0 1 0 0 0 4 1 1 5 0 0 R = , RN = = 1 1 2 E 1 1 A B RMR = = (1) 1 1 CD A = [ 1], m1 = 1, m2 = 1, m3 = 0. (1) Same for congruence T A is congruent to [ 1] J2 and congruent to [1] J2: [i][ 1][i] = [+1] ICTP School: Linear Algebra: Friday, June 26, 2009 11 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Example 2 1 0 1 MN 1 0 1 A = 0 1 0 = ; M = , N = 2 3 [0 0] 0 0 1 0 0 0 0 4 0 1 5 0 0 R = , RN = = 1 0 1 E 1 0 A B RMR = = (1) 0 1 CD A(1) = [1], m1 = 1, m2 = 1, m3 = 0. Same for congruence T A is both congruent and congruent to [1] J2. ICTP School: Linear Algebra: Friday, June 26, 2009 12 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Examples 1 & 2 The matrices in Examples 1 and 2 0 1 1 1 0 1 A = 0 0 1 and B = 0 1 0 2 0 0 0 3 2 0 0 0 3 4 5 4 5 are congruent since they are both congruent to [1] J2. However, they are not congruent: A is congruent to [ 1] J2, B is congruent to [+1] J2, and [s][ 1][s¯] = s 2 [ 1] = [+1] j j 6 ICTP School: Linear Algebra: Friday, June 26, 2009 13 Roger Horn (University of Utah) Matrix Canonical Forms / 22 The regular part 1 * De…ne: A := A . The matrix AA is the cosquare of A. If A S AS, then ! 1 1 AA S AS (S AS) = S AA S ! If A A S 1 (A A) S, then ! 1 1 AA S AS S A S = (S AS) (S AS) ! *congruence of A corresponds to similarity of AA The Jordan Canonical Form of the *cosquare of A is a *congruence invariant of A. T Same for congruence and cosquares A A ICTP School: Linear Algebra: Friday, June 26, 2009 14 Roger Horn (University of Utah) Matrix Canonical Forms / 22 The JCF of *cosquares and cosquares 1 1 s 1 s AA = A A AA = AA A A The Jordan Canonical Form of AA is a direct sum of blocks of two types: Jk (µ) 0 with 0 < µ < 1, and Jk (λ) with λ = 1 0 Jk (1/µ¯) j j j j 1 T T 1 T s T 1 T s T A A = A A A A = A A A A T The Jordan Canonical Form of A A is a direct sum of blocks of two types: Jk (µ) 0 k+1 k+1 with 0 = µ = ( 1) , and Jk (( 1) ) 0 Jk (1/µ) 6 6 ICTP School: Linear Algebra: Friday, June 26, 2009 15 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Canonical blocks for *congruence and congruence 2 1 1 3 Γk = (real) 6 1 1 7 6 7 6 1 1 7 6 7 6 1 1 7 6 7k k 4 5 0 1 Γ = [1] , Γ = 1 2 1 1 T k+1 Γk Γk is similar to Jk (( 1) ), so Γk is indecomposable under congruence or *congruence. ICTP School: Linear Algebra: Friday, June 26, 2009 16 Roger Horn (University of Utah) Matrix Canonical Forms / 22 Canonical blocks for *congruence and congruence 1 i ∆k = 2 3 (symmetric) 1 6 1 i 7 6 7k k 4 5 0 1 ∆ = [1] , ∆ = 1 2 1 i ∆k∆k is similar to Jk (1), so ∆k is indecomposable under *congruence.
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