Even if all singular values are nonzero, if then the cokernel is nontrivial, in which case is padded with orthogonal vectors from the cokernel.
12.
The private key is a basis B of a lattice L with good properties ( such as short nearly orthogonal vectors ) and a unimodular matrix U.
13.
The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors ( length of the chain ) is recorded.
14.
In general any two "'orthogonal states "'can be used, where an orthogonal vector pair is formally defined as one having a zero inner product.
15.
It is the built-in symmetry of the Cartesian coordinate system where coordinates are orthogonal vectors of unit length in flat Euclidean space that allows this generalization to apply so broadly.
16.
(where q \ geq 2 is an arbitrary natural number, and normalized means that the length of the vector is 1; usually s and n are orthogonal vectors ).
17.
To see that this is true, notice that the rows of " H " are all orthogonal vectors over the field of real numbers and each have length \ sqrt n.
18.
:P . S . A simple proof, but not aesthetically pleasing, is to take your " n " orthogonal vectors and place them into the columns of a matrix.
19.
The product "'e "'123 yields the bivector that is Hodge dual to in three dimensions, as discussed above, while "'e "'4 as a product of orthogonal vectors is also bivector valued.
20.
The cross product does not generalize to dimensions other than 3 as a product of two vectors, yielding a unique third orthogonal vector, and in higher dimensions not all " k "-vectors can be identified with vectors or scalars.
