- #1

- 234

- 6

## Main Question or Discussion Point

I think it's 3...

All 2x2 can be written as

[tex]a_1 A_1 + a_2 A_2 + a_3 A_3 + a_4 A_4[/tex]

with

[itex]A_1 =

\begin{bmatrix}

1 & 0 \\

0 & 0

\end{bmatrix}

[/itex], [itex]A_2 =

\begin{bmatrix}

0 & 1 \\

0 & 0

\end{bmatrix}

[/itex], [itex]A_3 =

\begin{bmatrix}

0 & 0 \\

1 & 0

\end{bmatrix}

[/itex], [itex]A_4 =

\begin{bmatrix}

0 & 0 \\

0 & 1

\end{bmatrix}

[/itex]

And 2x2 Symm = [itex]a_1 A1 + a_2 (A_2 + A_3) + a_4 A_4[/itex], and if we combine [itex]A_2 + A_3[/itex] into a single basis element [itex]A^*[/itex], then [itex]A^*[/itex] is still independent of [itex]A_1[/itex] and [itex]A_4[/itex]...so actually all 2x2 symm matrices should be in a space of dimension 3...and the basis elements are [itex]A_1 \ A_2 \ and \ A^*[/itex]?

All 2x2 can be written as

[tex]a_1 A_1 + a_2 A_2 + a_3 A_3 + a_4 A_4[/tex]

with

[itex]A_1 =

\begin{bmatrix}

1 & 0 \\

0 & 0

\end{bmatrix}

[/itex], [itex]A_2 =

\begin{bmatrix}

0 & 1 \\

0 & 0

\end{bmatrix}

[/itex], [itex]A_3 =

\begin{bmatrix}

0 & 0 \\

1 & 0

\end{bmatrix}

[/itex], [itex]A_4 =

\begin{bmatrix}

0 & 0 \\

0 & 1

\end{bmatrix}

[/itex]

And 2x2 Symm = [itex]a_1 A1 + a_2 (A_2 + A_3) + a_4 A_4[/itex], and if we combine [itex]A_2 + A_3[/itex] into a single basis element [itex]A^*[/itex], then [itex]A^*[/itex] is still independent of [itex]A_1[/itex] and [itex]A_4[/itex]...so actually all 2x2 symm matrices should be in a space of dimension 3...and the basis elements are [itex]A_1 \ A_2 \ and \ A^*[/itex]?