On Incidence Energy of Graphs
... concept of energy to all (not necessarily square) matrices, defining the energy of a matrix M as the sum of the singular values of M . Recall that the singular values of a matrix M are equal to the square roots of the eigenvalues of the (square) matrix M Mt . Let G = (V, E) be a simple graph, where V ...
... concept of energy to all (not necessarily square) matrices, defining the energy of a matrix M as the sum of the singular values of M . Recall that the singular values of a matrix M are equal to the square roots of the eigenvalues of the (square) matrix M Mt . Let G = (V, E) be a simple graph, where V ...
Null-Controllability of Linear Systems on Time Scales
... and matrix is regressive, then it is locally null -controllable. Proof: If R ∈ is arbitrary, then there exists a control B = − ' S' E , UBGAHI E , G, B ∈ , R ] ' , such that staW S R te x can be steered to 0 in a finite time. Since map → S , is rd-continuous, t ...
... and matrix is regressive, then it is locally null -controllable. Proof: If R ∈ is arbitrary, then there exists a control B = − ' S' E , UBGAHI E , G, B ∈ , R ] ' , such that staW S R te x can be steered to 0 in a finite time. Since map → S , is rd-continuous, t ...
Homework assignment, Feb. 18, 2004. Solutions
... Proof. Let dim V = r and let v1 , v2 , . . . , vr be a basis in V . Then the system of vectors Av1 , Av2 , . . . , Avr isa generating system for AV . Indeed, every vector v ∈ V can be αk vk , so any vector of form Av, v ∈ V can be represented as a represented as v = rk=1 linear combination Av = k ...
... Proof. Let dim V = r and let v1 , v2 , . . . , vr be a basis in V . Then the system of vectors Av1 , Av2 , . . . , Avr isa generating system for AV . Indeed, every vector v ∈ V can be αk vk , so any vector of form Av, v ∈ V can be represented as a represented as v = rk=1 linear combination Av = k ...
Higher Order GSVD for Comparison of Global mRNA Expression
... Figure 1. Higher-order generalized singular value decomposition (HO GSVD). In this raster display of Equation (1) with overexpression (red), no change in expression (black), and underexpression (green) centered at gene- and array-invariant expression, the S. pombe, S. cerevisiae and human global mRN ...
... Figure 1. Higher-order generalized singular value decomposition (HO GSVD). In this raster display of Equation (1) with overexpression (red), no change in expression (black), and underexpression (green) centered at gene- and array-invariant expression, the S. pombe, S. cerevisiae and human global mRN ...
A NON CONCENTRATION ESTIMATE FOR RANDOM MATRIX
... continues to hold for local fields of positive characteristic. The proof however will have to go back to more foundational results in random matrix products, which are not available to date. In particular, due to the lack of complete reducibility of reductive groups in characteristic p, one needs to ...
... continues to hold for local fields of positive characteristic. The proof however will have to go back to more foundational results in random matrix products, which are not available to date. In particular, due to the lack of complete reducibility of reductive groups in characteristic p, one needs to ...
A NOTE ON DERIVATIONS OF COMMUTATIVE ALGEBRAS 1199
... Our result (Theorem 1) seems useful in classifying simple algebras of degree one which satisfy identities giving rise to derivations of the algebras. For example, an immediate corollary to Theorem 1 is Kleinfeld and Kokoris' determination of simple flexible algebras of degree one [6]. Then in Theore ...
... Our result (Theorem 1) seems useful in classifying simple algebras of degree one which satisfy identities giving rise to derivations of the algebras. For example, an immediate corollary to Theorem 1 is Kleinfeld and Kokoris' determination of simple flexible algebras of degree one [6]. Then in Theore ...
Matrices and RRE Form Notation. R is the real numbers, C is the
... Theorem 0.6. Suppose that A is the (augmented) matrix of a linear system of equations, and B is obtained from A by a sequence of elementary row operations. Then the solutions to the system of linear equations corresponding to A and the system of linear equations corresponding to B are the same. To p ...
... Theorem 0.6. Suppose that A is the (augmented) matrix of a linear system of equations, and B is obtained from A by a sequence of elementary row operations. Then the solutions to the system of linear equations corresponding to A and the system of linear equations corresponding to B are the same. To p ...
Singular values of products of random matrices and polynomial
... transformation of polynomial ensembles. The result may be stated as follows: suppose X is a random matrix whose squared singular values form a polynomial ensemble and that G is a complex Ginibre matrix. Then the squared singular values of Y = GX are also a polynomial ensemble. See Theorem 2.1 below ...
... transformation of polynomial ensembles. The result may be stated as follows: suppose X is a random matrix whose squared singular values form a polynomial ensemble and that G is a complex Ginibre matrix. Then the squared singular values of Y = GX are also a polynomial ensemble. See Theorem 2.1 below ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.