Quaternion product as a matrix product

Pick a quaternion

p = p₀ + p₁i + p₂j + p₃k

and consider the function that acts on quaternions by multiplying them on the left by p.

If we think of q as a vector in R⁴ then this is a linear function of q, and so it can be represented by multiplication by a 4 * 4 matrix M_p.

It turns out

How might you remember or derive this matrix? Consider the matrix on the left below. It's easier to see the pattern here than in M_p.

You can derive M_p from this matrix.

Let's look at the second row, for example. The second row of M_p, when multiplied by q as a column vector, produces the i component of the product.

How do you get an i term in the product? By multiplying the i component of p by the real component of q, or by multiplying the real component of p times the i component of p, or by multiplying the i/ j component of p by the j component of q, or by multiplying the i/k component of p by the k component of q.

The other rows follow the same pattern. To get the x component of the product, you add up the products of the x/y term of p and the y term of q. Here x and y range over

{1, i, j, k}.

To get M_p from the matrix on the right, replace 1 with the real component of p, replace i with the i component of p, etc.

As a final note, notice that the off-diagonal elements of M_p are anti-symmetric:

m_ij = -m_ji

unless i = j.

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