Algebraic Topology · Problem 27

Cohomology — Allen Hatcher

Mathematics — Topology · skew-symmetric-bilinear-forms

Problem

Show that after a suitable change of basis, a skew-symmetric nonsingular bilinear form over

\mathbb{Z}

can be represented by a matrix consisting of

2 \times 2

blocks

\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

along the diagonal and zeros elsewhere. [For the matrix of a bilinear form, the following operation can be realized by a change of basis: Add an integer multiple of the

i^{th}

row to the

j^{th}

row and add the same integer multiple of the

i^{th}

column to the

j^{th}

column. Use this to fix up each column in turn. Note that a skew-symmetric matrix must have zeros on the diagonal.]

Related theory

Symmetric And Skew Symmetric Matrices

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