The Unapologetic Mathematician

Mathematics for the interested outsider

The Special Linear Group (and others)

We’ve got down the notion of the general linear group \mathrm{GL}(V) of a vector space V, including the particular case of the matrix group \mathrm{GL}(n,\mathbb{F}) of the space \mathbb{F}^n. We also have defined the orthogonal group \mathbb{O}(n,\mathbb{F}) of n\times n matrices over \mathbb{F} whose transpose and inverse are the same, which is related to the orthogonal group \mathrm{O}(V,B) of orthogonal transformations of the real vector space V preserving a specified bilinear form B. Lastly, we’ve defined the group \mathrm{U}(n) of unitary transformations on \mathbb{C}^nn\times n complex matrices whose conjugate transpose and inverse are the same.

For all of these matrix groups — which are all subgroups of some appropriate \mathrm{GL}(n,\mathbb{F}) — we have a homomorphism to the multiplicative group of \mathbb{F} given by the determinant. We originally defined the determinant on \mathrm{GL}(n\mathbb{F}) itself, but we can easily restrict it to any subgroup. We actually know that for unitary and orthogonal transformations the image of this homomorphism must lie in a particular subgroup of \mathbb{F}^\times. But in any case, the homomorphism must have a kernel, and this kernel turns out to be important.

In the case of the general linear group \mathrm{GL}(V), the kernel of the determinant homomorphism consists of the automorphisms of V with determinant {1}. We call this subgroup of \mathrm{GL}(V) the “special linear group” \mathrm{SL}(V), and transformations in this subgroup are sometimes called “special linear transformations”. Of course, we also have the particular special linear group \mathrm{SL}(n,\mathbb{F})\subseteq\mathrm{GL}(n,\mathbb{F}). When we take the kernel of any of the other groups, we prepend the adjective “special” and an \mathrm{S} to the notation. Thus we have the special orthogonal groups \mathrm{SO}(V,B) and \mathrm{SO}(n,\mathbb{F}) and the special unitary group \mathrm{SU}(n).

In a sense, all the interesting part of the general linear group is contained in the special linear subgroup. Outside of that, what remains is “just” a scaling. It’s a little more complicated than it seems on the surface, but not much.

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September 8, 2009 - Posted by | Algebra, Linear Algebra

6 Comments »

  1. [...] Shears Generate the Special Linear Group We established that if we restrict to upper shears we can generate all upper-unipotent matrices. On the other hand if we use all shears and scalings we can generate any invertible matrix we want (since swaps can be built from shears and scalings). We clearly can’t build any matrix whatsoever from shears alone, since every shear has determinant and so must any product of shears. But it turns out that we can use shears to generate any matrix of determinant — those in the special linear group. [...]

    Pingback by Shears Generate the Special Linear Group « The Unapologetic Mathematician | September 9, 2009 | Reply

  2. [...] shears alone generate the special linear group. Can we strip them down any further? And, with this in mind, how many generators does it take to [...]

    Pingback by How Many Generators? « The Unapologetic Mathematician | September 11, 2009 | Reply

  3. [...] this gives a geometric meaning to the special orthogonal group . Orthogonal transformations send orthonormal bases to other orthonormal bases, which will send [...]

    Pingback by Parallelepipeds and Volumes II « The Unapologetic Mathematician | November 3, 2009 | Reply

  4. [...] to turn our heads and translate the laws of physics to compensate exactly. These rotations form the special orthogonal group of orientation- and inner product-preserving transformations, but we can also throw in reflections [...]

    Pingback by The Cross Product and Pseudovectors « The Unapologetic Mathematician | November 10, 2009 | Reply

  5. Sorry for commenting OT … what WP theme do you use? It’s looking great!!

    Comment by Sleemameameva | December 7, 2009 | Reply

  6. /me checks the bottom of the page

    “Theme: Andreas04 by Andreas Viklund.”

    Comment by John Armstrong | December 7, 2009 | Reply


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