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The following statements are equivalent: (i) A is positive definite. (ii) A can be factored as A diagonal entries. - RTR where R is an upper triangular matrix with positive x Ax> 0 for all nonzero x ER".

The following statements are equivalent: (i) A is positive definite. (ii) A can be factored as A diagonal entries. - RTR where R is an upper triangular matrix with positive x Ax> 0 for all nonzero x ER".

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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Definition A matrix A E R"x" is said to be positive definite if it is symmetric and has an LU factorization in which each pivot is positive. The following statements are equivalent: (i) A is positive definite. A can be factored as A diagonal entries. Ax> 0 for all nonzero xER". - RTR where R is an upper triangular matrix with positive Prove the equiv ences (i) <--> (ii), (ii) <---> (iii), (i) <--> (iii) Hint: The following theorems may hep: Let A € R¹x¹ be an arbitrary matrix such that y¹Cy > 0 for all nonzero vectors y E R¹ Thm 1: The matrix A is nonsingular Thm 2: For k = 1,...,n, the leading principal submatrix Ak satisfies g¹Akg > 0 for all nonzero vectors g € R¹, and hence each matrix Ak is also nonsingular
Transcribed Image Text:Definition A matrix A E R"x" is said to be positive definite if it is symmetric and has an LU factorization in which each pivot is positive. The following statements are equivalent: (i) A is positive definite. A can be factored as A diagonal entries. Ax> 0 for all nonzero xER". - RTR where R is an upper triangular matrix with positive Prove the equiv ences (i) <--> (ii), (ii) <---> (iii), (i) <--> (iii) Hint: The following theorems may hep: Let A € R¹x¹ be an arbitrary matrix such that y¹Cy > 0 for all nonzero vectors y E R¹ Thm 1: The matrix A is nonsingular Thm 2: For k = 1,...,n, the leading principal submatrix Ak satisfies g¹Akg > 0 for all nonzero vectors g € R¹, and hence each matrix Ak is also nonsingular
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