Exercise 3.5. Let H be a symmetric matrix with spectral decomposition H = V DV". (a) Show that an eigenvector v associated with a positive eigenvalue A satisfies v™Hv > 0. (b) If H is nonsingular, write down the inverse of H in terms of V and D. (c) If r is a positive integer, give an expression for H" in terms of D and V. If H is positive definite, find a matrix B such that H = B² = BB (B is the “square root" of H).

Exercise 3.5. Let H be a symmetric matrix with spectral decomposition H = V DV". (a) Show that an eigenvector v associated with a positive eigenvalue A satisfies v™Hv > 0. (b) If H is nonsingular, write down the inverse of H in terms of V and D. (c) If r is a positive integer, give an expression for H" in terms of D and V. If H is positive definite, find a matrix B such that H = B² = BB (B is the “square root" of H).

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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Exercise 3.5. Let H be a symmetric matrix with spectral decomposition H = V DVT.
(a) Show that an eigenvector v associated with a positive eigenvalue d satisfies v"Hv > 0.
(b) If H is nonsingular, write down the inverse of H in terms of V and D.
(c) If r is a positive integer, give an expression for H" in terms of D and V. If H is positive
definite, find a matrix B such that H = B² = BB (B is the "square root" of H).

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