question about： The S + N Decomposition-Linear Systems and Canonical Forms of Operators 1. For each of the following operators T find bases for the generalized eigenspaces;give the matrices (for the standard basis) of the semisimple and nilpotentparts of T.(a)[]

In order to find the generalized eigenspaces, we first need to find the eigenvalues of the matrix T.…

Answered: 1. For each of the following operators…

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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* 2: The function f(x) = x² + 2x² + 3x² + 4x₁x₂ + 5x₂x3 + 6x1x3 can be written in the form f(X) = XTQX with the matrix [1 2 3 [1 2 31 1 3 A: Q = 2 B: Q=4 5 6 C: Q = 2 5 3 D: Non of these 2 2.5 2.5 3 13 L2 3 1. 5 4 A B О с D 126
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can you explain part 3b fully please
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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1. For each of the following operators T find bases for the generalized eigenspaces; give the matrices (for the standard basis) of the semisimple and nilpotent parts of T. (a) []