# Book: CartanMatrix

Interpret Geometrically and Combinatorially the Relationship Between a Root System and Its Cartan Matrix

A generalized Cartan matrix is a square matrix {$A = (a_{i j})$}:

• 1) with integral entries
• 2) such that {$(a_{ii})=2$},
• 3) for non-diagonal entries, {$a_{ij} \leq 0$},
• 4a) {$A$} can be written as {$D S$}, where {$D$} is a diagonal matrix, and {$S$} is a symmetric matrix.
• 4b) {$A$} is positive definite, that is, it satisfies Sylvester's criterion.
• 4c) The volume spanned by any subset of roots is positive in orientation.
• 4d) The roots have positive projections onto the same vector.
• 5) Consequently, {$a_{ij}=0$} if and only if {$a_{ji}=0$}.

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 positive-definite.

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 non-zero 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 {$M$}

• {$M$} defines a multiset of eigenvectors that reflect the extent to which the matrix is nondegenerate. The eigenvectors are a natural coordinate system underlying a matrix.
• A change of basis changes the eigenvectors, but this can be reversed.
• The action of a matrix is given by its eigenvalues, and this is not affected by a change of basis.
• If {$M$} is positive definite, then the eigenvectors are mutually orthogonal, and can be taken to be an orthonormal basis. The matrix thus determines a coordinate system.
• {$M$} is positive definite if and only if the eigenvalues are all positive, real numbers.
• The effect of the matrix is to stretch vectors in accordance with the coordinate system. There are no rotations.

Theorems

• If {$A$} is Hermitian, there exists an orthonormal basis of {$V$} consisting of eigenvectors of {$A$}. Each eigenvalue is real.

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.

• {$M$} is positive definite if and only if all of its eigenvalues are positive.