I Basis of 2x2 matrices with real entries

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What is the basis of 2x2 matrices with real entries? I know that the basis of 2x2 matrices with complex entries are 3 Pauli matrices and unit matrix:
\begin{bmatrix}<br /> 0 &amp; 1 \\[0.3em]<br /> 1 &amp; 0 \\[0.3em]<br /> \end{bmatrix},
\begin{bmatrix}<br /> 0 &amp; -i \\[0.3em]<br /> i &amp; 0 \\[0.3em]<br /> \end{bmatrix}
\begin{bmatrix}<br /> 1 &amp; 0 \\[0.3em]<br /> 0 &amp; -1 \\[0.3em]<br /> \end{bmatrix}
and
\begin{bmatrix}<br /> 1 &amp; 0 \\[0.3em]<br /> 0 &amp; 1 \\[0.3em]<br /> \end{bmatrix}
What about in the case of real 2x2 matrices? How many matrices is there in the basis?
 
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It's the same problem as selecting a basis of ##\mathbf{R}^4## e.g. via the correspondence ##\begin{pmatrix} a_1 & a_2 \\ a_3 & a_4 \end{pmatrix} \leftrightarrow (a_1,a_2,a_3,a_4)##
 
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So I again need four matrices? As in a complex case? Is the basis then
\begin{bmatrix}<br /> 1 &amp; 0 \\[0.3em]<br /> 0 &amp; 0 \\[0.3em]<br /> \end{bmatrix},
\begin{bmatrix}<br /> 0 &amp; 1 \\[0.3em]<br /> 0 &amp; 0 \\[0.3em]<br /> \end{bmatrix},
\begin{bmatrix}<br /> 0 &amp; 0 \\[0.3em]<br /> 1 &amp; 0 \\[0.3em]<br /> \end{bmatrix}
and
\begin{bmatrix}<br /> 0 &amp; 0 \\[0.3em]<br /> 0 &amp; 1 \\[0.3em]<br /> \end{bmatrix}?
 
That's fine, but it's of course not unique.
 
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Yes. Thank you. So I can see if some 2x2 matrix
\begin{bmatrix}<br /> a_{11} &amp; a_{12} \\[0.3em]<br /> a_{21} &amp; a_{22} \\[0.3em]<br /> \end{bmatrix}
commute with
\begin{bmatrix}<br /> 1 &amp; 0 \\[0.3em]<br /> 0 &amp; 0 \\[0.3em]<br /> \end{bmatrix}
and
\begin{bmatrix}<br /> 0 &amp; 1 \\[0.3em]<br /> 0 &amp; 0 \\[0.3em]<br /> \end{bmatrix}
and
\begin{bmatrix}<br /> 0 &amp; 0 \\[0.3em]<br /> 1 &amp; 0 \\[0.3em]<br /> \end{bmatrix}
and
\begin{bmatrix}<br /> 0 &amp; 0 \\[0.3em]<br /> 0 &amp; 1 \\[0.3em]<br /> \end{bmatrix}
then the matrix commute with any 2x2 matrix?
 
Well yes, but I think the only matrices that commute with all four of those are of the form ##\mathrm{diag}(a,a) = aI##, i.e. multiples of the identity.
 
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LagrangeEuler said:
Yes. Thank you. So I can see if some 2x2 matrix
\begin{bmatrix}<br /> a_{11} &amp; a_{12} \\[0.3em]<br /> a_{21} &amp; a_{22} \\[0.3em]<br /> \end{bmatrix}
commute with
\begin{bmatrix}<br /> 1 &amp; 0 \\[0.3em]<br /> 0 &amp; 0 \\[0.3em]<br /> \end{bmatrix}
and
\begin{bmatrix}<br /> 0 &amp; 1 \\[0.3em]<br /> 0 &amp; 0 \\[0.3em]<br /> \end{bmatrix}
and
\begin{bmatrix}<br /> 0 &amp; 0 \\[0.3em]<br /> 1 &amp; 0 \\[0.3em]<br /> \end{bmatrix}
and
\begin{bmatrix}<br /> 0 &amp; 0 \\[0.3em]<br /> 0 &amp; 1 \\[0.3em]<br /> \end{bmatrix}
then the matrix commute with any 2x2 matrix?
Since you're asking about a basis for ##\mathbb M_{2\times2}##, square matrices of order 2 with real entries, commutivity doesn't enter into things. For a basis, you need four linearly independent matrices ##M_1, M_2, M_3, M_4## that span ##\mathbb M_{2\times2}##. The matrices you list in post 5 are the standard basis for this space, although as already stated, there are other possibilities for a basis.
 
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ergospherical said:
Well yes, but I think the only matrices that commute with all four of those are of the form ##\mathrm{diag}(a,a) = aI##, i.e. multiples of the identity.
Yes of course. It is also a center of group ##GL_2(\mathbb{R})##.
 
Mark44 said:
Since you're asking about a basis for ##\mathbb M_{2\times2}##, square matrices of order 2 with real entries, commutivity doesn't enter into things. For a basis, you need four linearly independent matrices ##M_1, M_2, M_3, M_4## that span ##\mathbb M_{2\times2}##. The matrices you list in post 5 are the standard basis for this space, although as already stated, there are other possibilities for a basis.
Ok. But using that you can for instance to find the center of group ##GL_2(\mathbb{R})##.
 
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LagrangeEuler said:
So I again need four matrices? As in a complex case?

The Pauli matrices plus the identity are not a basis for all complex 2x2 matrices. They are a basis for the Hermitian 2x2 complex matrices.

To cover all 2x2 complex matrices you need a basis with 8 elements.
 
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Really I am pretty sure that I read that those matrices are basis in ##\mathbb{C}^{2x2}##.
 
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The Bill said:
The Pauli matrices plus the identity are not a basis for all complex 2x2 matrices. They are a basis for the Hermitian 2x2 complex matrices.

To cover all 2x2 complex matrices you need a basis with 8 elements.
LagrangeEuler said:
Really I am pretty sure that I read that those matrices are basis in ##\mathbb{C}^{2x2}##.
This all depends whether the field of scalars are the Real or Complex numbers. Complex 2x2 matrices can be seen as a 4D Vector Space over ##\mathbb C##, and the set of Hermitian matrices is not a subspace. Or, they can be seen as an 8D Vector Space over ##\mathbb R##, in which case the set of Hermitian matrices is a 4D subspace.
 
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