- #1
kostoglotov
- 234
- 6
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]?