Thread: Finding a basis

1. Finding a basis

Let V be the real vector space of all 3×3 hermitian matrices with complex entries. Find a basis for V.

2. Originally Posted by iheartmathrookie
Let V be the real vector space of all 3×3 hermitian matrices with complex entries. Find a basis for V.
The real vector space? Then a basis consists of $\displaystyle \begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}i & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}0 & i & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, etc. for a total of 18 "basis" matrices.

3. Let $\displaystyle E_{ij}$ be the matrix with a 1 in the ij position and 0 elsewhere.

Can you show

$\displaystyle E_{11}$ , $\displaystyle E_{22}$ , $\displaystyle E_{33}$

$\displaystyle E_{12}+ E_{21}$

$\displaystyle E_{13} + E_{31}$

$\displaystyle E_{23} + E_{32}$

$\displaystyle iE_{12}-iE_{21}$

$\displaystyle iE_{13}-iE_{31}$

$\displaystyle iE_{23}-iE_{32}$

form a basis?

4. Originally Posted by HallsofIvy
The real vector space? Then a basis consists of $\displaystyle \begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}i & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}0 & i & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, etc. for a total of 18 "basis" matrices.

Hermitian matrices have real diagonal entries and equal their transpose conjugates, so with this restriction we are left with a (9 dimensional) subspace of the real 3 x 3 matrices.

5. Right. For some reason, my eyes passed right over the word "Hermitian"!