Choose one from he following list of inner products and then find the adjoint of:
$$ \left[
\begin{array}{ c c }
1 & 2 \\
-1 & 3
\end{array} \right]
$$
When your inner prod cut is used on both its domain and target space.
a. The Euclidean dot product
b. The weighted inner product $\langle v,w\rangle=2v_1w_1+3v_1w_1$
I guess I don't really understand how to find the domain or target space of a matrix.
All I could find online was that the adjoint is the transpose of the cofactor matrix but I'm not sure how to calculate the cofactor matrix.
Any help is greatly appreciated!
Best Answer
If you have an inner product $<\cdot, \cdot>$, the adjoint of an operator $A$ is $A^*$ where $<Ax,y> = < x,A^*y>$ for any $x,y$.
For the dot product, $(Ax) \cdot y = (A x)^T y = x^T A^T y = x^T (A^T y) = x \cdot (A^T y)$ so the adjoint of $A$ is $A^T$.
Now, try the other inner product on your own.