What is the algebra generated by a set of matrices?

Question

From Wikipedia:

A set of matrices $A_1, \ldots, A_k$ are said to be simultaneously triangularizable if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix $P$.

Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the $A_i$, denoted $K[A_1,\ldots,A_k]$. Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a Borel subalgebra."

I do not understand the second paragraph. What does "the algebra of matrices it generates, namely all polynomials in the $A_i$, denoted $K[A_1,\ldots,A_k]$" mean?

Answer 1

If you have two matrices $A$ and $B$, you can do many things with them. You can add them ($A+B$), you can multiply them ($AB$), you can take powers ($A^n$). When you are allowed to do addition and multiplication and these two actions play nicely with eachother (ie $a(b+c)=ab+ac$, etc..) you say that your set (or ring or whatever you started with that you found out now has both addition and multiplication) is an algebra.

The algebra generated by some matrices $A_i$ can be described as "all the polynomials on the $A_i$'s". This is nothing more that a rephrasing of what we said above -we just call $A_1^2A_2+A_3A_1^4$ a polynomial on the matrices $A_1,A_2,A_3$.

The finer point is being made in the next sentence, where it says that simultaneous triangularizability (god, what a word...) implies this algebra can be made to live in a particularly nice algebra called "the Lie subalgebra of upper-triangular matrices". It also gives you a characterization of when some matrices are simultaneously triangularizable, ie when the algebra they generate is equivalent to some nice generic type of algebra: "a Lie subalgebra of a Borel subalgebra".

For more, I believe it would be nice to look at a few wikipedia pages, like the one on Lie Algebras and the one on Borel subalgebras.