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Book: CartanMatrix

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.


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


Given a matrix {$M$}


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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Puslapis paskutinį kartą pakeistas 2019 birželio 09 d., 13:03