A Topological Approach to Left Eigenvalues of Quaternionic Matrices M.J. Pereira-S´aezy (joint work with E. Mac´ıas-Virg´os∗) yDepartamento de Econom´ıaAplicada II, Universidade da Coru~na ∗Departamento de Xeometr´ıae Topolox´ıa, Universidade de Santiago de Compostela Sevilla, September 9-14, 2012 1 Left eigenvalues Notion Study's determinant Characteristic map 2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra 3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 2 / 69 Left eigenvalues 1 Left eigenvalues Notion Study's determinant Characteristic map 2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra 3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 3 / 69 Left eigenvalues Notion 1 Left eigenvalues Notion Study's determinant Characteristic map 2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra 3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 4 / 69 Left eigenvalues Notion Notion Definition Let A 2 Mn×n(H), a quaternion λ 2 H is said to be a left eigenvalue of the matrix A if Av = λv for some vector v 2 Hn; v 6= 0: Equivalently, A − λI is a singular matrix. For a matrix A 2 Mn×n(H), we denote σl (A) its left spectrum, i.e. the set of left eigenvalues. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 5 / 69 Huang and So [Linear Algebra Appl., 2001] proved that a 2 × 2 matrix may have one, two or infinite left eigenvalues. A different proof was presented by EMV and MJPS [Electron. J. Linear Algebra, 2009]. Both proofs are algebraic in nature and seemingly difficult to generalize for n > 2. Left eigenvalues Notion About the existence of left eigenvalues: Wood [Bull. Lond. Math. Soc., 1985] proved that any n × n quaternionic matrix A has at least one left eigenvalue. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 6 / 69 A different proof was presented by EMV and MJPS [Electron. J. Linear Algebra, 2009]. Both proofs are algebraic in nature and seemingly difficult to generalize for n > 2. Left eigenvalues Notion About the existence of left eigenvalues: Wood [Bull. Lond. Math. Soc., 1985] proved that any n × n quaternionic matrix A has at least one left eigenvalue. Huang and So [Linear Algebra Appl., 2001] proved that a 2 × 2 matrix may have one, two or infinite left eigenvalues. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 6 / 69 Both proofs are algebraic in nature and seemingly difficult to generalize for n > 2. Left eigenvalues Notion About the existence of left eigenvalues: Wood [Bull. Lond. Math. Soc., 1985] proved that any n × n quaternionic matrix A has at least one left eigenvalue. Huang and So [Linear Algebra Appl., 2001] proved that a 2 × 2 matrix may have one, two or infinite left eigenvalues. A different proof was presented by EMV and MJPS [Electron. J. Linear Algebra, 2009]. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 6 / 69 Left eigenvalues Notion About the existence of left eigenvalues: Wood [Bull. Lond. Math. Soc., 1985] proved that any n × n quaternionic matrix A has at least one left eigenvalue. Huang and So [Linear Algebra Appl., 2001] proved that a 2 × 2 matrix may have one, two or infinite left eigenvalues. A different proof was presented by EMV and MJPS [Electron. J. Linear Algebra, 2009]. Both proofs are algebraic in nature and seemingly difficult to generalize for n > 2. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 6 / 69 * λ 2 H is on σl (A) iff the matrix A − λI is not invertible. Left eigenvalues Notion { There is no systematic study of left eigenvalues for n > 2. { In this talk we try a topological approach that could be generalized to other dimensions. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 7 / 69 Left eigenvalues Notion { There is no systematic study of left eigenvalues for n > 2. { In this talk we try a topological approach that could be generalized to other dimensions. * λ 2 H is on σl (A) iff the matrix A − λI is not invertible. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 7 / 69 Left eigenvalues Study's determinant 1 Left eigenvalues Notion Study's determinant Characteristic map 2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra 3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 8 / 69 Left eigenvalues Study's determinant Study's determinant It is possible to generalize to the quaternions the norm j det j (that is, with real values) of the complex determinant. Definition Let the quaternionic matrix A 2 Mn×n(H) be decomposed as A = X + jY with X ; Y 2 Mn×n(C). We shall call Study's determinant of A the non-negative real number Sdet(A) := (det c(A))1=2; X −Y where c(A) is the complex matrix c(A) = 2 M ( ): Y X 2n×2n C M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 9 / 69 Left eigenvalues Study's determinant Proposition Study's determinant Sdet is the only functional that verifies the following properties : 1 Sdet(AB) = Sdet(A) · Sdet(B); 2 if A is a complex matrix then Sdet(A) = j det(A)j. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 10 / 69 Left eigenvalues Study's determinant The following immediate consequences are very useful for computations. Corollary 1 Sdet(A) > 0 if and only if the matrix A is invertible; −1 2 let A and B = PAP be similar matrices, then Sdet(A) = Sdet(B); 3 Sdet(A) does not change when a (right) multiple of one column is added to another column. 4 Sdet(A) does not change when a (left) multiple of one row is added to another row; 5 Sdet(A) does not change when two columns (or two rows) are permuted. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 11 / 69 Left eigenvalues Study's determinant We shall need the following property too Proposition For any matrix formed by two boxes A; B of order m and n respectively, it holds that A 0 Sdet = Sdet(A) · Sdet(B): ∗ B M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 12 / 69 Left eigenvalues Characteristic map 1 Left eigenvalues Notion Study's determinant Characteristic map 2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra 3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 13 / 69 Proposition The spectrum σl (A) is compact. Left eigenvalues Characteristic map Characteristic map Definition The map µ : H ! H is a characteristic map of the matrix A 2 Mn×n(H) if, up to a constant, its norm verifies jµ(λ)j = Sdet(A − λI) for all λ 2 H. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 14 / 69 Left eigenvalues Characteristic map Characteristic map Definition The map µ : H ! H is a characteristic map of the matrix A 2 Mn×n(H) if, up to a constant, its norm verifies jµ(λ)j = Sdet(A − λI) for all λ 2 H. Proposition The spectrum σl (A) is compact. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 14 / 69 Left eigenvalues Characteristic map For example, let D = diag(q1;:::; qn) be a diagonal matrix. Then µ(λ) = (q1 − λ) ··· (qn − λ) is a characteristic map for D. For a triangular matrix it is the same. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 15 / 69 Spectrum of 2 × 2 matrices 1 Left eigenvalues Notion Study's determinant Characteristic map 2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra 3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 16 / 69 Spectrum of 2 × 2 matrices Remark The left spectrum is not invariant by similarity. However, if P is an invertible real matrix then Sdet(A − λI) = Sdet(PAP−1 − λI), hence A and PAP−1 have the same characteristic maps. M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 17 / 69 if bc = 0 (triangular matrix), σl (A) = fa; dg; µ(λ) = (d − λ)(a − λ): we study now the bc 6= 0 case.