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A simple question about matrix groups. According to Geometrical Methods of Mathematical Physics (Schutz, p. 95),

The one-parameter subgroup generated by any matrix $A$ is the integral curve through [the identity] $e$ of the left-invariant vector field whose tangent at $e$ is $A$.

The author is referring to the general linear group in $n$ real dimensions, $GL(n,\mathrm{R})$.

So for $GL(n,\mathrm{R})$, the tangent vectors at $e$ are represented by matrices. But any vector $\overline{V}$ can be written in terms of a basis, say $$\overline{V}= V^1\frac{\partial}{\partial x^1}+V^2\frac{\partial}{\partial x^2} + ...$$

So how do we express a matrix, such as $\begin{bmatrix} a & b\\ c & d\\ \end{bmatrix}$, in terms of basis vectors $\frac{\partial}{\partial x^i}$? Wouldn't we need to express it with a rank-2 basis: $\frac{\partial}{\partial x^i}\otimes\frac{\partial}{\partial x^j}$?

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  • $\begingroup$ You do not use $\partial /\partial x^i$ as a basis in the space of matrices. Such a basis is given by matrices where one matrix entry equals $1$ and the rest are zero. For instance, the space of 2-by-2 matrices is 4-dimensional, its basis consists of four elements. $\endgroup$ – Moishe Kohan Oct 8 '17 at 5:14
  • $\begingroup$ The key point about the basis vectors $\partial/\partial x^i$ is that we know how they transform with respect to a coordinate change, e.g., $\frac{\partial}{\partial x^i} = \frac{\partial q^j}{\partial x^i}\frac{\partial}{\partial q^j}$ - this is what makes vectors a geometrical object, not just an array of numbers. So if the tangent vector for the matrix group is a geometrical object, how does it transform? $\endgroup$ – Doubt Oct 8 '17 at 23:08
  • $\begingroup$ Transforms under what? Where do you want to do a coordinate change? It is a standard linear algebra exercise to work out coordinate change for column-vectors under a linear transformation. And why do you regard this as a "key point"? (As a separate example, think about the Hilbert space $L^2([0,1])$, whose elements are functions regarded as vectors. What would be the "key point" in this case?) $\endgroup$ – Moishe Kohan Oct 9 '17 at 21:12

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