Quaternions as 4x4 Matrices - Connections to Linear Algebra

In math, it's usually possible to view an object or concept from many different (but equivalent) angles. In this video, we will see that the quaternions may be viewed as 4x4 real-valued matrices of a special form. What is interesting here is that if you know how to multiply matrices, you already know how to do more or less any calculation with quaternions. Moreover, we will see that the fundamental relations between the quaternionic imaginary units, i, j, and k also hold in matrix form.

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[vivekdabholkar]

Many thanks, makes it very easy to follow without excessive effort. This is how one should be taught to retain infomation!

[SaadTaameOfficial]

Awesome explanation. Just a little thing that you might want to create a video to explain. There are two matrices that we can extract from a quaternion multiplication p * q. We can extract the matrix for p (which corresponds to multiplying by a quaternion on the left) and we can extract the matrix for q (which corresponds to multiplying by a quaternion on the right). Fastforward to the video where you describe 3D rotations by using quaternions: here the formula is given by q * v * conj(q). Now people who write code are often presented with a routine to convert from quaternion to rotation matrix. This can be done by extracting the matrix for q (call it A) and extracting the matrix for conj(q) (call it B) and then multiplying B by A (or A by B). The matrices commute in this case. I hope that I did not make mistakes.

[feedbackex]

Thank you so much, this is best explanation I have seen.

[mohammadhashemi]

Some awesome explanations here! Thank you for this amazing video, really helful

[cybernaut_ev]

Very insightful. Have never seen it done like this before.

[lohdiwei]

Certainly the most tiresome array of ones and zeroes and minuses that I have ever -- with a wee bit of admiration -- awarded a thumbs-up to!
👌😊

[giobrach]

In other words, there is a ring isomorphism between the ring of quaternions and a subring of 4x4 real antisymmetric matrices. There is also a similar isomorphism between the ring of quaternions and a subring of 2x2 *complex* matrices: through it, the four quaternionic units are translated into the Pauli matrices. To see this, divide the 4x4 matrices corresponding to the units into 2x2 blocks and "reverse" the usual isomorphism between complex numbers and 2x2 real antisymmetric matrices with identical elements on the diagonal ("2D rotation matrices")

[brucemoore]

Great Videos. Question for you though. When you represented a quaternion as a 4x4 matrix in the beginning of this video, why did represent only one of the quaternions as a 4x4, the other as a 4x1, and not both matrices as 4x4's?

Thank You in advanced for you response

bruceam

[david_porthouse]

On the dodgy computer that I use, the CPU adds on small random errors whenever it does a multiplication. This will be disastrous if the graphics system uses vectors and t-matrices to do a chain of rotations. However with quaternions we can continually renormalise newly-generated quaternions so it isn’t so bad. If you replace quaternions with 4x4 matrices, well yes you can do it, but you are throwing away a principal merit of quaternions.

[jimnewton]

If a matrix times a matrix equal the matrix times a vector, then I think of eigenvalues and eigenvectors. What's the relation of quaternions to eigenvectors?

[stephenmorais]

Is there a way to do qvq-1 using matrices?

[SandburgNounouRs]

How can you multiply a (1, 4) matrix (vector) by another (1, 4) matrix and have for result a vector (1, 4), the dimensions are incompatible, right ?

[gideonbuckwalter]

Cool, so 1, i, j, and k are their own inverses!