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See: Four Classical Algebras, Four Classical Algebras Draft, Classical Lie groups, Lie theory Interpret Geometrically and Combinatorially the Relationship Between a Root System and Its Cartan Matrix A generalized Cartan matrix is a square matrix {$A = (a_{i j})$}:
Define the vector onto which the roots have positive projections. This means that there is a global notion of positivity which coordinates the notions of positivity in each dimension. Geometrically interpret and relate the vectors defined by the rows of the Cartan matrix to the roots. {$$a_{ji}=\frac{2(r_i,r_j)}{(r_j,r_j)}$$} Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positivedefinite. A symmetric {$n \times n$} real matrix {$M$} is said to be positive definite if the scalar {$z^{\textsf {T}}Mz$} is strictly positive for every nonzero column vector {$z$} of {$n$} real numbers. In other words, {$$\sum_{i,j}^{n}z_iM_{ij}z_j > 0$$} This means that the vectors {$z$} and {$Mz$} are always positive projections upon each other, that is, they are always on the same side of a hyperplane. Furthermore, the fact that {$M$} is symmetric means that this inner product can be transposed and the value stays the same. Equivalently, the leading principal minors of {$M$} are all positive, including the determinant. Geometrically, this means that the volume of the matrix can be understood as given by row vectors that are nondegenerate and make for a positive volume. When the matrix is reduced to upper triangular form, all of the diagonal terms are positive. Thus the determinants are positive. Thus there is a basis such that the matrix Observations Given a matrix {$M$}
Theorems
The matrix representation of {$A$} in a basis of eigenvectors is diagonal, and by the construction the proof gives a basis of mutually orthogonal eigenvectors; by choosing them to be unit vectors one obtains an orthonormal basis of eigenvectors.
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