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3. A Cartan matrix A is a square matrix whose elements aij satisfy the following conditions: 1. aij is an integer, one of { –3, –2, –1,0, 2 } 2. ajj = 2 for all diagonal elements of A 3. aij <0 off of the diagonal 4. aij = 0 iff aji = 0 5. There exists an invertible diagonal matrix D such that DAD-1 gives a symmetric and positive definite quadratic form. Give a 2 x 2 non-diagonal Cartan matrix and its eigenvalues.

3. A Cartan matrix A is a square matrix whose elements aij satisfy the following conditions: 1. aij is an integer, one of { –3, –2, –1,0, 2 } 2. ajj = 2 for all diagonal elements of A 3. aij <0 off of the diagonal 4. aij = 0 iff aji = 0 5. There exists an invertible diagonal matrix D such that DAD-1 gives a symmetric and positive definite quadratic form. Give a 2 x 2 non-diagonal Cartan matrix and its eigenvalues.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. A Cartan matrix A is a square matrix whose elements aij satisfy the following conditions: 1. aij is an integer, one of {-3, –2, – 1,0,2 } 2. ajj = 2 for all diagonal elements of A 3. aij <0 off of the diagonal 4. aij = 0 iff aji = 0 5. There exists an invertible diagonal matrix D such that DAD-1 gives a symmetric and positive definite quadratic form. Give a 2 x 2 non-diagonal Cartan matrix and its eigenvalues.
Transcribed Image Text:3. A Cartan matrix A is a square matrix whose elements aij satisfy the following conditions: 1. aij is an integer, one of {-3, –2, – 1,0,2 } 2. ajj = 2 for all diagonal elements of A 3. aij <0 off of the diagonal 4. aij = 0 iff aji = 0 5. There exists an invertible diagonal matrix D such that DAD-1 gives a symmetric and positive definite quadratic form. Give a 2 x 2 non-diagonal Cartan matrix and its eigenvalues.
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