Steiner Equiangular Tight Frames Redux
... on the other hand, E(b, v) = 1, we know that the bth block of the design contains the vth point. For example, for the (4, 2, 1)-BIBD whose incidence matrix E given in (3), the fact that E(5, 4) = 1 implies that the fifth block of the design contains the fourth point. Now recall that there are K poin ...
... on the other hand, E(b, v) = 1, we know that the bth block of the design contains the vth point. For example, for the (4, 2, 1)-BIBD whose incidence matrix E given in (3), the fact that E(5, 4) = 1 implies that the fifth block of the design contains the fourth point. Now recall that there are K poin ...
xi. linear algebra
... This demonstrates that we need a new way to define positive and negative for matrices, at least as far as second-derivative tests are concerned. Definition: An n ! n matrix A is positive semidefinite if for all vectors x !R n , the number x ! Ax " 0 . Definition: An n ! n matrix A is negative semid ...
... This demonstrates that we need a new way to define positive and negative for matrices, at least as far as second-derivative tests are concerned. Definition: An n ! n matrix A is positive semidefinite if for all vectors x !R n , the number x ! Ax " 0 . Definition: An n ! n matrix A is negative semid ...
4.3 COORDINATES IN A LINEAR SPACE By introducing
... 4.3 COORDINATES IN A LINEAR SPACE By introducing coordinates, we can transform any n-dimensional linear space into Rn 4.3.1 Coordinates in a linear space Consider a linear space V with a basis B consisting of f1, f2, ...fn. Then any element f of V can be written uniquely as f = c1f1 + c2f2 + · · · + ...
... 4.3 COORDINATES IN A LINEAR SPACE By introducing coordinates, we can transform any n-dimensional linear space into Rn 4.3.1 Coordinates in a linear space Consider a linear space V with a basis B consisting of f1, f2, ...fn. Then any element f of V can be written uniquely as f = c1f1 + c2f2 + · · · + ...
TEST I Name___________________________________ Show
... 12) A chemistry department wants to make 3 liters of a 17.5% basic solution by mixing 20% solution with a 15% solution. How many liters of each type of basic solution should be used to produce the 17.5% solution? ...
... 12) A chemistry department wants to make 3 liters of a 17.5% basic solution by mixing 20% solution with a 15% solution. How many liters of each type of basic solution should be used to produce the 17.5% solution? ...
1 SPECIALIS MATHEMATICS - VECTORS ON TI 89
... Alternatively use your Vectors program to find the angle, magnitudes, dot product etc. ...
... Alternatively use your Vectors program to find the angle, magnitudes, dot product etc. ...
Math 4707: Introduction to Combinatorics and Graph Theory
... repeated use of a slight generalization of this method results in the Laplace expansion of a matrix, but we will not need this for our purposes.) We use these properties, in particular, multilinearity of the determinant, to prove the Matrix-Tree-Theorem described above. First, we need to prove anoth ...
... repeated use of a slight generalization of this method results in the Laplace expansion of a matrix, but we will not need this for our purposes.) We use these properties, in particular, multilinearity of the determinant, to prove the Matrix-Tree-Theorem described above. First, we need to prove anoth ...
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.