Finding a basis

**iheartmathrookie** · September 11th 2009, 11:17 AM

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

**HallsofIvy** · September 11th 2009, 02:41 PM

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 $\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$ , $\begin{bmatrix}i & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$ , $\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$ , $\begin{bmatrix}0 & i & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$ , etc. for a total of 18 "basis" matrices.

**siclar** · September 11th 2009, 02:47 PM

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

Can you show

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

$E_{12}+ E_{21}$

$E_{13} + E_{31}$

$E_{23} + E_{32}$

$iE_{12}-iE_{21}$

$iE_{13}-iE_{31}$

$iE_{23}-iE_{32}$

form a basis?

**siclar** · September 11th 2009, 02:50 PM

Originally Posted by HallsofIvy

The real vector space? Then a basis consists of $\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$ , $\begin{bmatrix}i & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$ , $\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$ , $\begin{bmatrix}0 & i & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$ , 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.

**HallsofIvy** · September 12th 2009, 03:47 AM

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

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