Lie Algebras
... somewhat longer but completely straightforward calculation shows that if A is an m × n matrix and B is an n × k matrix, then t (AB) = t B t A. The dual space of a vector space V is L(V, F) (where F is viewed as a onedimensional vector space), and is denoted V ∗ . Its elements are called linear funct ...
... somewhat longer but completely straightforward calculation shows that if A is an m × n matrix and B is an n × k matrix, then t (AB) = t B t A. The dual space of a vector space V is L(V, F) (where F is viewed as a onedimensional vector space), and is denoted V ∗ . Its elements are called linear funct ...
TRANSLATION FUNCTORS AND DECOMPOSITION NUMBERS
... modules is different from the combinatorics of costandard modules. In particular, the standard and costandard modules with the same highest weight have different dimensions, (Lemma 3.1.1). We call them therefore thick respectively thin Kac modules. A remarkable observation (Lemma 3.2.1) is the exist ...
... modules is different from the combinatorics of costandard modules. In particular, the standard and costandard modules with the same highest weight have different dimensions, (Lemma 3.1.1). We call them therefore thick respectively thin Kac modules. A remarkable observation (Lemma 3.2.1) is the exist ...
A PRIMER ON CARNOT GROUPS
... definitions from the algebraic viewpoint, with some basic properties. In particular, we show the uniqueness of stratifications in Section 1.2. In Section 2 we introduce the homogeneous distances and the general definition of Carnot-Carathéodory spaces. In Section 2.5 we show the continuity of homoge ...
... definitions from the algebraic viewpoint, with some basic properties. In particular, we show the uniqueness of stratifications in Section 1.2. In Section 2 we introduce the homogeneous distances and the general definition of Carnot-Carathéodory spaces. In Section 2.5 we show the continuity of homoge ...
Invariant Theory of Finite Groups
... Finally, we need to show that we have a Groebner basis. In Exercise 12, we will ask you to prove that k LT(gk ) = x k · This is where we use lex order with x1 > · · · > xn > y1 > · · · > yn . Thus the leading terms of g1 , . . . , gk are relatively prime, and using the theory developed in §9 of Chap ...
... Finally, we need to show that we have a Groebner basis. In Exercise 12, we will ask you to prove that k LT(gk ) = x k · This is where we use lex order with x1 > · · · > xn > y1 > · · · > yn . Thus the leading terms of g1 , . . . , gk are relatively prime, and using the theory developed in §9 of Chap ...
REAL ALGEBRAIC GEOMETRY. A FEW BASICS. DRAFT FOR A
... 1. Real affine (and projective!) algebraic varieties We start from a complex (algebraic) variety X ⊂ Cn+1 (affine case) or X ⊂ PnC (projective case). Proposition 1.1. The following properties for X are equivalent: (i) the ideal I(X) has real generators f1 , . . . , fk ∈ R[x0 , . . . , xn ]. (ii) X = ...
... 1. Real affine (and projective!) algebraic varieties We start from a complex (algebraic) variety X ⊂ Cn+1 (affine case) or X ⊂ PnC (projective case). Proposition 1.1. The following properties for X are equivalent: (i) the ideal I(X) has real generators f1 , . . . , fk ∈ R[x0 , . . . , xn ]. (ii) X = ...
Lie Algebras and Representation Theory
... As an exercise in that direction, the reader should try to describe the orbits of the action of GL(n, R) on Rn × Rn defined by A · (x, y) := (Ax, Ay). 1.2. Group representations. A representation of a group G is simply an action of G on a K–vector space V by linear maps. So we need an action ϕ : G × ...
... As an exercise in that direction, the reader should try to describe the orbits of the action of GL(n, R) on Rn × Rn defined by A · (x, y) := (Ax, Ay). 1.2. Group representations. A representation of a group G is simply an action of G on a K–vector space V by linear maps. So we need an action ϕ : G × ...
Local structure of generalized complex manifolds
... V is defined either as an R-linear automorphism preserves J of V ⊕V , which the standard symmetric bilinear pairing ·, · and satisfies J 2 = −1, or as a complex subspace L ⊂ VC ⊕ VC∗ , which is isotropic with respect to the C-bilinear extension of ·, · and satisfies VC ⊕ VC∗ = L ⊕ L. There is no ...
... V is defined either as an R-linear automorphism preserves J of V ⊕V , which the standard symmetric bilinear pairing ·, · and satisfies J 2 = −1, or as a complex subspace L ⊂ VC ⊕ VC∗ , which is isotropic with respect to the C-bilinear extension of ·, · and satisfies VC ⊕ VC∗ = L ⊕ L. There is no ...
chap4.pdf
... Matrix rank: number of linearly independent column (row) vectors For 2 × 2 matrix columns define an [a1 , a2 ]-system Full rank=2: a1 and a2 are linearly independent Rank deficient: matrix that does not have full rank If a1 and a2 are linearly dependent then matrix has rank 1 Also called a singular ...
... Matrix rank: number of linearly independent column (row) vectors For 2 × 2 matrix columns define an [a1 , a2 ]-system Full rank=2: a1 and a2 are linearly independent Rank deficient: matrix that does not have full rank If a1 and a2 are linearly dependent then matrix has rank 1 Also called a singular ...
1. Algebra of Matrices
... • A is a square matrix (of order n), if m = n; the n-ple (a11 , . . . , ann ) is called diagonal of A. The set Mn,n (R) will be simply denoted by Mn (R). • A square matrix A = (aij ) ∈ Mn (R) is said upper triangular [resp. lower triangular] if aij = 0, for all i > j [resp. if aij = 0, for all i < j ...
... • A is a square matrix (of order n), if m = n; the n-ple (a11 , . . . , ann ) is called diagonal of A. The set Mn,n (R) will be simply denoted by Mn (R). • A square matrix A = (aij ) ∈ Mn (R) is said upper triangular [resp. lower triangular] if aij = 0, for all i > j [resp. if aij = 0, for all i < j ...
Lie Groups and Lie Algebras
... This definition is more general than what we will use in the course, where we will restrict ourselves to so-called matrix Lie groups. The manifold will then always be realised as a subset of some Rd . For example the manifold S 3 , the three-dimensional sphere, can be realised as a subset of R4 by t ...
... This definition is more general than what we will use in the course, where we will restrict ourselves to so-called matrix Lie groups. The manifold will then always be realised as a subset of some Rd . For example the manifold S 3 , the three-dimensional sphere, can be realised as a subset of R4 by t ...
Geometric Means - College of William and Mary
... To establish the validity of this definition we must show that the limit does indeed exist and that it has the asserted form. We do this in the next theorem. (r) ...
... To establish the validity of this definition we must show that the limit does indeed exist and that it has the asserted form. We do this in the next theorem. (r) ...
Research Article Computing the Square Roots of a Class of
... 3.3. Square Roots of Hermitian k-Circulant Matrices In fact, if G−1 BG B, then A B2 implies G−1 AG A. That means Hermitian k-circulant matrices may have square roots which are still Hermitian k-circulant. Although it is unknown whether it is true, if it has a Hermitian k-circulant square root, ...
... 3.3. Square Roots of Hermitian k-Circulant Matrices In fact, if G−1 BG B, then A B2 implies G−1 AG A. That means Hermitian k-circulant matrices may have square roots which are still Hermitian k-circulant. Although it is unknown whether it is true, if it has a Hermitian k-circulant square root, ...
A geometric proof of the Berger Holonomy Theorem
... speaking, negative scalar curvature (by the Ricci identity). The normal curvature tensor at p is regarded as a linear map from Λ2 (Tp M ) to Λ2 (νp (M )). Then one has to average R̃ over the normal holonomy group and apply the Cartan construction of symmetric spaces, in order to obtain that the norm ...
... speaking, negative scalar curvature (by the Ricci identity). The normal curvature tensor at p is regarded as a linear map from Λ2 (Tp M ) to Λ2 (νp (M )). Then one has to average R̃ over the normal holonomy group and apply the Cartan construction of symmetric spaces, in order to obtain that the norm ...
![arXiv:math/0607084v3 [math.NT] 26 Sep 2008](http://s1.studyres.com/store/data/014838781_1-63e86760b3063903e159a8155f16fef1-300x300.png)
arXiv:math/0607084v3 [math.NT] 26 Sep 2008
... para a Ciência e a Tecnologia” (FCT), cofinanced by the European Community Fund FEDER/POCTI. ...
... para a Ciência e a Tecnologia” (FCT), cofinanced by the European Community Fund FEDER/POCTI. ...
Algebraically positive matrices - Server
... Proof. Let A be an algebraically positive matrix and let f be a real polynomial such that f (A) > 0. Let λ1 , . . . , λn be the eigenvalues of A. The spectral mapping theorem [7, p.8] asserts that the eigenvalues of f (A) are f (λ1 ), . . . , f (λn ). Since f (A) is a positive matrix, the Perron-Fro ...
... Proof. Let A be an algebraically positive matrix and let f be a real polynomial such that f (A) > 0. Let λ1 , . . . , λn be the eigenvalues of A. The spectral mapping theorem [7, p.8] asserts that the eigenvalues of f (A) are f (λ1 ), . . . , f (λn ). Since f (A) is a positive matrix, the Perron-Fro ...
Square Roots of-1 in Real Clifford Algebras
... quaternions and by Grassmann’s exterior algebra. Grassmann invented the antisymmetric outer product of vectors, that regards the oriented parallelogram area spanned by two vectors as a new type of number, commonly called bivector. The bivector represents its own plane, because outer products with ve ...
... quaternions and by Grassmann’s exterior algebra. Grassmann invented the antisymmetric outer product of vectors, that regards the oriented parallelogram area spanned by two vectors as a new type of number, commonly called bivector. The bivector represents its own plane, because outer products with ve ...
Matrix Decomposition and its Application in Statistics
... Theorem: If A is a n×n real, symmetric and positive definite matrix then there exists a unique lower triangular matrix G with positive diagonal element such that A GG T . Proof: Since A is a n×n real and positive definite so it has a LU decomposition, A=LU. Also let the lower triangular matrix L t ...
... Theorem: If A is a n×n real, symmetric and positive definite matrix then there exists a unique lower triangular matrix G with positive diagonal element such that A GG T . Proof: Since A is a n×n real and positive definite so it has a LU decomposition, A=LU. Also let the lower triangular matrix L t ...
Notes on Classical Groups - School of Mathematical Sciences
... Multiplication by zero induces the zero endomorphism of (F, +). Multiplication by any non-zero element induces an automorphism (whose inverse is multiplication by the inverse element). In particular, we see that the automorphism group of (F, +) acts transitively on its non-zero elements. So all non ...
... Multiplication by zero induces the zero endomorphism of (F, +). Multiplication by any non-zero element induces an automorphism (whose inverse is multiplication by the inverse element). In particular, we see that the automorphism group of (F, +) acts transitively on its non-zero elements. So all non ...
MATH08007 Linear Algebra S2, 2011/12 Lecture 1
... λ is a real or complex scalar. For this to make sense, we need to be able to add our objects X and Y and multiply them by scalars. The first key idea we shall meet is that of sets with a linear structure, that is sets in which there are natural definitions of addition and scalar multiplication. Simp ...
... λ is a real or complex scalar. For this to make sense, we need to be able to add our objects X and Y and multiply them by scalars. The first key idea we shall meet is that of sets with a linear structure, that is sets in which there are natural definitions of addition and scalar multiplication. Simp ...
here - The Institute of Mathematical Sciences
... These notes were initially prepared for a summer school in the MTTS programme more than a decade ago. They are intended to introduce an undergraduate student to the basic notions of linear algebra, and to advocate a geometric rather than a coordinate-dependent purely algebraic approach. Thus, the go ...
... These notes were initially prepared for a summer school in the MTTS programme more than a decade ago. They are intended to introduce an undergraduate student to the basic notions of linear algebra, and to advocate a geometric rather than a coordinate-dependent purely algebraic approach. Thus, the go ...
Universal Identities I
... (2.1) implies a similar formula for sums of three squares in any commutative ring by specializing the 6 indeterminates to any 6 elements of any commutative ring. So (2.1) implies that sums of three squares are closed under multiplication in any commutative ring, but this false in Z: 3 and 5 are sums ...
... (2.1) implies a similar formula for sums of three squares in any commutative ring by specializing the 6 indeterminates to any 6 elements of any commutative ring. So (2.1) implies that sums of three squares are closed under multiplication in any commutative ring, but this false in Z: 3 and 5 are sums ...
Sec 3 Add Maths : Matrices
... The interior design company is given the job of putting up the curtains for the windows, sliding doors and the living room of the entire new apartment block of the NTUC executive condominium. There are a total of 156 threebedroom units and each unit has 5 windows, 3 sliding doors and 2 living rooms. ...
... The interior design company is given the job of putting up the curtains for the windows, sliding doors and the living room of the entire new apartment block of the NTUC executive condominium. There are a total of 156 threebedroom units and each unit has 5 windows, 3 sliding doors and 2 living rooms. ...
7 Eigenvalues and Eigenvectors
... Definition 7.9 We say A is triangularizable if there exists an invertible matrix C such that C −1 AC is upper triangular. Remark 7.7 Obviously all diagonalizable matrices are triangularizable. The following result says that triangularizability causes least problem: Proposition 7.9 Over the complex n ...
... Definition 7.9 We say A is triangularizable if there exists an invertible matrix C such that C −1 AC is upper triangular. Remark 7.7 Obviously all diagonalizable matrices are triangularizable. The following result says that triangularizability causes least problem: Proposition 7.9 Over the complex n ...
chap9.pdf
... Subspace u = sv + tw has dimension 2 since it is spanned by two vectors These vectors have to be non-collinear — otherwise they define a line, or a 1D subspace Example: orthogonal projection of w onto v — projection lives in 1D subspace formed by v If vectors v, w collinear — called linearly depende ...
... Subspace u = sv + tw has dimension 2 since it is spanned by two vectors These vectors have to be non-collinear — otherwise they define a line, or a 1D subspace Example: orthogonal projection of w onto v — projection lives in 1D subspace formed by v If vectors v, w collinear — called linearly depende ...