diagonalForm -- the Grothendieck-Witt class of a diagonal form

Synopsis

• Usage:

  diagonalForm(k,a)

  diagonalForm(k,L)

• Inputs:
  • k, a ring, a field
  • a, a ring element, any element $a\in k$
  • L, a sequence, of elements $a_{i} \in k$, where $i = 1,\dots, n$
• Outputs:
  • a Grothendieck Witt Class, the diagonal form $\langle a_1,\ldots,a_n\rangle \in \text{GW}(k)$

Description

Given a sequence of elements $a_1,\ldots,a_n \in k$ we can form the diagonal form $\langle a_1,\ldots,a_n\rangle$ defined to be the block sum of each of the rank one forms $\langle a_i \rangle \colon k \times k \to k$ $(x,y) \mapsto a_i xy$.

i1 : diagonalForm(QQ,(3,5,7))

o1 = GrothendieckWittClass{cache => CacheTable{}}
                           matrix => | 3 0 0 |
                                     | 0 5 0 |
                                     | 0 0 7 |

o1 : GrothendieckWittClass

Inputting a ring element, an integer, or a rational instead of a sequence will produce a rank one form instead. For instance:

i2 : diagonalForm(GF(29),5/13)

o2 = GrothendieckWittClass{cache => CacheTable{}}
                           matrix => | -13 |

o2 : GrothendieckWittClass

i3 : diagonalForm(RR,2)

o3 = GrothendieckWittClass{cache => CacheTable{}}
                           matrix => | 2 |

o3 : GrothendieckWittClass

See also

• diagonalClass -- produces a diagonalized form for any Grothendieck-Witt class, with simplified terms on the diagonal
• congruenceDiagonalize -- diagonalizing a symmetric matrix via congruence
• diagonalEntries -- extracts a list of diagonal entries for a GrothendieckWittClass