Skew-coninvolutory matrices

Abara, Ma. Nerissa M.; Merino, Dennis I.; Paras, Agnes T.

doi:10.1016/j.laa.2007.05.037

Cited by 10 publications

(4 citation statements)

References 2 publications

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“…If λ k = 1, let T be an involution such that T (J k (λ) ⊕ [1]) is similar to J k (1) ⊕ [−1]. Observe that T ⊕ T T ∈ SI 2k+2 and that rank((T ⊕ T T )P − I) ≥ 2k, and so (T ⊕ T T )P is similar to (…”

Section: Proof By Lemma 4(a) It Is Enough To Consider E(n)mentioning

confidence: 99%

“…If k = 2, let a 2 = λ 2 and T be an involution such that T (J 2 (λ) ⊕ [1]) is similar to diag(1, a, −a). Observe that T P is symplectically similar to J 2 (1) (diag(a, −a, a −1 , −a −1 )) or I 2 (diag(a, −a, a −1 , −a −1 )), which by Lemmas 11 and 19, are products of three symplectic involutions.…”

Section: If λ K = −1 Let T Be An Involution Such That T (J K (λ) ⊕ [mentioning

confidence: 99%

“…Observe that T P is symplectically similar to J 2 (1) (diag(a, −a, a −1 , −a −1 )) or I 2 (diag(a, −a, a −1 , −a −1 )), which by Lemmas 11 and 19, are products of three symplectic involutions. If k = 3, let a 2 = λ 3 and T be an involution such that T (J 3 (λ) ⊕ [1]) is similar to diag (i, −i, a, −a). Then T P is symplectically similar , −a, a −1 , −a −1 ) or diag(i, i, a, −a, −i, −i, a −1 , −a −1 ), which by Theorem 8 and Lemmas 11 and 17 are both products of three symplectic involutions.…”

Section: If λ K = −1 Let T Be An Involution Such That T (J K (λ) ⊕ [mentioning

confidence: 99%

“…Then T P is symplectically similar , −a, a −1 , −a −1 ) or diag(i, i, a, −a, −i, −i, a −1 , −a −1 ), which by Theorem 8 and Lemmas 11 and 17 are both products of three symplectic involutions. If k > 3, let T be an involution such that T (J k (λ) ⊕ [1]) is similar to J k−1 (1) ⊕diag(a, −a). Lemma 26.…”

Section: If λ K = −1 Let T Be An Involution Such That T (J K (λ) ⊕ [mentioning

confidence: 99%

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Each symplectic matrix is a product of four symplectic involutions

Cruz

2015

Linear Algebra and its Applications

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Available online xxxx Submitted by V.V. Sergeichuk MSC: 15A21 15A23 Keywords: Symplectic matrices Involutions Coninvolutions Square zero matrices Hamiltonian matricesGustafson, Halmos, and Radjavi in 1973 proved that each matrix A with det A = ±1 is a product of four involutions. We prove that these involutions can be taken to be symplectic if A is symplectic (every symplectic matrix has unit determinant). Using this result we give an alternative proof of Laffey's theorem that every nonsingular even size matrix is a product of skew symmetric matrices. Ballantine in 1978 proved that each matrix A with | det A| = 1 is a product of four coninvolutions. We prove that these coninvolutions can be taken to be symplectic if A is symplectic. We also prove that each Hamiltonian matrix is a sum of two square zero Hamiltonian matrices.

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Section: Proof By Lemma 4(a) It Is Enough To Consider E(n)mentioning

confidence: 99%

Section: If λ K = −1 Let T Be An Involution Such That T (J K (λ) ⊕ [mentioning

confidence: 99%

Section: If λ K = −1 Let T Be An Involution Such That T (J K (λ) ⊕ [mentioning

confidence: 99%

Section: If λ K = −1 Let T Be An Involution Such That T (J K (λ) ⊕ [mentioning

confidence: 99%

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