eigenmatrix

[AntonioLao]

The theory of eigenvalues (l) and eigenvectors (X) is well investigated in mathematics. In certain respects the search of these eigenvalues boils down to solving the proper characteristic polynomial equations if and only if the determinant of the difference between the product of these eigenvalues with the identity matrix and a given square matrix (A) vanishes, det(lI – AX) = 0, where I is the identity matrix. This is a general expression that applies to any rank of the square matrices: A and I. Expressed as eigenvectors, the expression becomes AX = lX, where X, by general agreements, are column vectors. For the remaining brief discussions, these remarks served as introduction to the question of existence of eigenvalues when the eigenvectors are generalized to eigenmatrices: AH = lH, where H are Hadamard matrices.

The special properties of these matrices reduce the number of possible solutions to two, namely H+ and H- and for the trivial case; the matrix A is either H+ or H- and the l take on positive integer values. Asides from their unique properties, Hadamard matrices look like tensors of rank greater than or equal to two.

[mkirkpatrick]

The theory of eigenvalues (l) and eigenvectors (X) is well investigated in mathematics. In certain respects the search of these eigenvalues boils down to solving the proper characteristic polynomial equations if and only if the determinant of the difference between the product of these eigenvalues with the identity matrix and a given square matrix (A) vanishes, det(lI – AX) = 0, where I is the identity matrix. This is a general expression that applies to any rank of the square matrices: A and I. Expressed as eigenvectors, the expression becomes AX = lX, where X, by general agreements, are column vectors. For the remaining brief discussions, these remarks served as introduction to the question of existence of eigenvalues when the eigenvectors are generalized to eigenmatrices: AH = lH, where H are Hadamard matrices.

The special properties of these matrices reduce the number of possible solutions to two, namely H+ and H- and for the trivial case; the matrix A is either H+ or H- and the l take on positive integer values. Asides from their unique properties, Hadamard matrices look like tensors of rank greater than or equal to two.

H+ and H- are these two negitive,and positive,there seems to be a relationship here
with the two primal forces,Ying and Yang,and I feel also that there has to be an applied
third,which some how balances out the two in question.

regards michael.

[AntonioLao]

are these two negitive,and positive

They are matrices. The problem with matrices is that there are no negative or positive indicating multidimensional directions.

[mkirkpatrick]

They are matrices. The problem with matrices is that there are no negative or positive indicating multidimensional directions.

That could then present a problem,how then would we bore deeper into energy,without
a positive or negative drill?





regards michael.

[AntonioLao]

a positive or negative drill?

This is only useful as far as unitary symmetry U(1) is intended. More symmetry is needed for SU(2) or SU(3) or even higher symmetry.

[mkirkpatrick]

This is only useful as far as unitary symmetry U(1) is intended. More symmetry is needed for SU(2) or SU(3) or even higher symmetry.

Right,will have to rethink this one then!


regards michael.