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# addGW -- the direct sum of two Grothendieck-Witt classes

- Usage:
    `addGW(beta1, beta2)`
- Inputs:
  - beta1, a Grothendieck-Witt Class, a Grothendieck-Witt class
  - beta2, a Grothendieck-Witt Class, a Grothendieck-Witt class
- Outputs:
  - gamma, a Grothendieck-Witt Class, a Grothendieck-Witt class representing the direct sum of the two input classes

## Description

Given two GrothendieckWittClass objects beta1 and beta2, this method computes the direct sum of the two classes, which is a new GrothendieckWittClass object representing the isomorphism class of the direct sum of the two symmetric bilinear forms represented by the input classes. The resulting class has a Gram matrix which is the direct sum of the Gram matrices of the two input classes.

    i1 : R = QQ[x]/(x^2 + 1);

    i2 : beta1 = makeGWClass matrix(R, {{1,2},{2,x}});

    i3 : beta2 = makeGWClass matrix(R, {{3,4},{4,5}});

    i4 : addGW(beta1, beta2)

    o4 = | 1 2 0 0 |
         | 2 x 0 0 |
         | 0 0 3 4 |
         | 0 0 4 5 |

    o4 : GrothendieckWittClass

## See also

- GrothendieckWittClass -- a new type intended to capture the isomorphism class of an element of the Grothendieck-Witt ring of a field or finite étale algebras over a field
- makeGWClass -- the Grothendieck-Witt class of a symmetric matrix
- multiplyGW -- the tensor product of two Grothendieck-Witt classes

## Ways to use addGW:

- `addGW(GrothendieckWittClass,GrothendieckWittClass)`

## For the programmer

The object addGW is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/A1BrouwerDegrees/Documentation/GrothendieckWittClassesDoc.m2:117:0.