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Antisymmetric Matrix

An Antisymmetric Matrix or Skew Symmetric Matrix is a matrix where its cross-diagonal entries are negated, that is

Aij=−Aji\boldsymbol{A}_{ij} = -\boldsymbol{A}_{ji}

Since

Aii=−Aii\boldsymbol{A}_{ii} = -\boldsymbol{A}_{ii}

it follows that the diagonal elements must be 00 ([📖Eve80, 31ff.]).

Examples​

Skew symmetric 4×44 \times 4 matrix​

Let

A=(0ab−c−a0e−h−b−e0−ichi0)\boldsymbol{A} = \begin{pmatrix} 0 & a & b & -c \\ -a & 0 & e & -h \\ -b & -e & 0 & -i \\ c & h & i & 0 \end{pmatrix}

be a matrix. Then A\boldsymbol{A} is skew symmetric, since

  • A01=a, A10=−a⇒A10=−A01\boldsymbol{A}_{01} = a,\ \boldsymbol{A}_{10} = -a \Rightarrow \boldsymbol{A}_{10} = -\boldsymbol{A}_{01}
  • A02=b, A20=−b⇒A20=−A02\boldsymbol{A}_{02} = b,\ \boldsymbol{A}_{20} = -b \Rightarrow \boldsymbol{A}_{20} = -\boldsymbol{A}_{02}
  • A03=−c, A30=c⇒A30=−A03\boldsymbol{A}_{03} = -c,\ \boldsymbol{A}_{30} = c \Rightarrow \boldsymbol{A}_{30} = -\boldsymbol{A}_{03}
  • A12=e, A21=−e⇒A21=−A12\boldsymbol{A}_{12} = e,\ \boldsymbol{A}_{21} = -e \Rightarrow \boldsymbol{A}_{21} = -\boldsymbol{A}_{12}
  • A13=−h, A31=h⇒A31=−A13\boldsymbol{A}_{13} = -h,\ \boldsymbol{A}_{31} = h \Rightarrow \boldsymbol{A}_{31} = -\boldsymbol{A}_{13}
  • A23=−i, A32=i⇒A32=−A23\boldsymbol{A}_{23} = -i,\ \boldsymbol{A}_{32} = i \Rightarrow \boldsymbol{A}_{32} = -\boldsymbol{A}_{23}
  • Diagonal: Aii=0\boldsymbol{A}_{ii} = 0

Cross-product as skew-symmetric matrix​

The cross-product of two vectors a⃗,b⃗∈R3\vec{a}, \vec{b} \in \mathbb{R}^3 can be written as the Matrix-Vector-product of a skew-symmetric matrix and a vector:

Let

a⃗=(axayaz),b⃗=(bxbybz)\vec{a} = \begin{pmatrix}a_x\\a_y\\a_z\end{pmatrix}, \vec{b} = \begin{pmatrix}b_x\\b_y\\b_z\end{pmatrix}

Then

(axayaz)×(bxbybz)=(aybz−azbyazbx−axbzaxby−aybx)=(0−azby+aybzazbx+0−axbz−aybx+axby+0)=(0−azayaz0−ax−ayax0)(bxbybz)\begin{pmatrix}a_x\\a_y\\a_z\end{pmatrix} \times \begin{pmatrix}b_x\\b_y\\b_z\end{pmatrix} = \begin{pmatrix}a_y b_z - a_z b_y\\a_z b_x - a_x b_z\\a_x b_y - a_y b_x\end{pmatrix} = \begin{pmatrix} 0 &- &a_z b_y &+ &a_y b_z \\ a_z b_x &+ &0 &-&a_x b_z \\ -a_y b_x &+ &a_x b_y &+ &0 \end{pmatrix} = \begin{pmatrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{pmatrix} \begin{pmatrix} b_x \\ b_y \\ b_z \end{pmatrix}

References

  1. [Eve80]: Eves, H.W.: Elementary Matrix Theory (1980), Dover [BibTeX]