I have a problem with understanding the concept of Left (Right) Linear Independence in Quaternion Matrix. I was studying the article 'Quaternions and Matrices of Quaternions*' by Fuzhen Zhang and I want to learn more about ranks of Quaternion Matrices.
As we know The rank of a quaternion matrix A is defined to be the maximum
number of columns of A which are right linearly independent. Here is an example from that article:
I did my calculations and the columns were left linearly independent and right linearly dependent, so inversely than in the article. Multiplication of quaternions is not alternate, so I understand that left linearly independent means that vectors are multiplied by the appropriate number (i, j, k) from the left side to satisfy the equation Ax = 0 (xA = 0 for right LI – Do I think right?) and we can check if they are left(right) linearly independent or dependent. Im taking here the definition:
The set ${x_{1}, . . . , x_n}$ is linearly independent if $c_1x_1 + · · · + c_nx_n = 0 ⇒ c_1=c_2= · · · =c_n=0.$
The qestion is: How should it looks like? And what is the best way to check the LI over the real ring of Quaternions? Maybe my way of checking the LI of this A matrix was wrong after all. I hope someone can explain it to me in a few simple words.
Best Answer
Left linear dependence means there is a nontrivial linear combination $$\require{cancel} q_1 a_1+q_2 a_2=0 $$ where $a_1,a_2$ are the columns of $A$, and $q_i\in\mathbb{H}$. It is easy to see that $q_1=i$ and $q_2=-1$ works: $$ i\begin{pmatrix}1\\j\end{pmatrix}-\begin{pmatrix}i\\k\end{pmatrix}=0. $$ Similarly, right linear dependence means $a_1q_1+a_2q_2=0$ for some nonzero $(q_1,q_2)\in\mathbb{H}^2$. To see that $a_1,a_2$ are right linear independent, note that the first component gives $q_1+iq_2=0$ or equivalently, $q_2=iq_1$. Then the second component reads $jq_1+\cancelto{j}{ki}q_1=0$, i.e., $2jq_1=0$ which implies $q_1=q_2=0$.