What is a skew-symmetric matrix?

A skew-symmetric matrix is a matrix representation of the vector cross product.

Let’s consider two 3D vectors:

\[\vec{v}_1 = (a_1, a_2, a_3) \quad \text{and} \quad \vec{v}_2 = (b_1, b_2, b_3)\]

The cross product (which results in a vector perpendicular to both vectors) is defined as:

\[\vec{v}_1 \times \vec{v}_2 = (a_2 b_3 - a_3 b_2, \quad a_3 b_1 - a_1 b_3, \quad a_1 b_2 - a_2 b_1)\]

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Skew-symmetric matrix form

The brilliant minds before us discovered a matrix form to encode this operation, called the skew-symmetric matrix:

\[[w]_\times = \begin{bmatrix} 0 & -w_z & w_y \\ w_z & 0 & -w_x \\ -w_y & w_x & 0 \end{bmatrix}\]

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Using the vector

\[\vec{v}_2 = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}\]

the matrix multiplication

\[[w]_\times \cdot \vec{v}_2 = \begin{bmatrix} 0 & -w_z & w_y \\ w_z & 0 & -w_x \\ -w_y & w_x & 0 \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = \begin{bmatrix} - w_z b_2 + w_y b_3 \\ w_z b_1 - w_x b_3 \\ - w_y b_1 + w_x b_2 \end{bmatrix}\]

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Example substitution

Substitute \(\vec{w} = (0, 1, 0)\):

\[[w]_\times \cdot \vec{v}_2 = \begin{bmatrix} b_3 \\ 0 \\ - b_1 \end{bmatrix}\]

ay, voila! This matches the vector cross product result:

\[\vec{w} \times \vec{v}_2 = (b_3, \; 0, \; -b_1)\]

So,

\[\mathbf{A}^T = \mathbf{B}\]

where \(\mathbf{A}\) is the matrix result and \(\mathbf{B}\) the vector notation.

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Notes on dimensions:

• Vector × Vector: Both must be 3D vectors (shape \(1 \times 3\) or \(3 \times 1\)) for cross product.
• Matrix × Vector: Number of matrix columns must equal vector rows (e.g., \(3 \times 3\) matrix times \(3 \times 1\) vector).

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