Collection of Maths Problems

Symmetric matrix of a form

Task number: 2738

• Variant 1

  For \(\mathbb K=\mathbb R\).

• Result

  \(f(\mathbf u,\mathbf v)=\mathbf u^T \mathbf A \mathbf v= u_1v_1 -2u_1v_2 +2u_2v_1-u_2v_3-2u_3v_1+2u_3v_2\).
  \(g(\mathbf v)=\mathbf v^T \mathbf A \mathbf v= v_1^2 - 2v_1v_3 + v_2v_3\).

  The same form yields the symmetric matrix \(\mathbf A'=\begin{pmatrix} 1 & 0 & -1 \\ 0 & 0 & \frac12 \\ -1 & \frac12 & 0 \\ \end{pmatrix} \).

• Variant 2

  For \(\mathbb K=\mathbb Z_2\) (the number 2 in \(\mathbb Z_2\) interpret as 0).

• Result

  \(f(\mathbf u,\mathbf v)=\mathbf u^T \mathbf A \mathbf v= u_1v_1+u_2v_3\).
  \(g(\mathbf v)=\mathbf v^T \mathbf A \mathbf v= v_1^2 +v_2v_3\).
  A symmetric matrix of this form does not exist.

• Variant 3

  For \(\mathbb K=\mathbb Z_3\).

• Result

  \(f(\mathbf u,\mathbf v)=\mathbf u^T \mathbf A \mathbf v= u_1v_1+u_1v_2 +2u_2v_1+2u_2v_3+u_3v_1+2u_3v_2\).
  \(g(\mathbf v)=\mathbf v^T \mathbf A \mathbf v= v_1^2 + v_1v_3 + v_2v_3\).

  The same form yields the symmetric matrix \(\mathbf A'=\begin{pmatrix} 1 & 0 & 2 \\ 0 & 0 & 2 \\ 2 & 2 & 0 \\ \end{pmatrix} \).