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1. For the following quadratic forms (i) write down the symmetric matrix A (so the form is equal to x"Ax), (ii) find matrices P, orthonormal, and D so that D diagonal, (iii) write the quadratic form in terms of y1,..., Yn where x = decide whether the form is positive/negative definite/semidefinite or indefinite. (a) 2x – 2x1x2 + 2x3 (b) –xỉ + 4x1x2 – 4x3 (c) 3x? + 4.x3 + 5xž + 4x1x2 – 4x2X3 (it may help to show that A characteristic polynomial of A). РТ АР is Py and (iv) 1 is a root of the %3D

1. For the following quadratic forms (i) write down the symmetric matrix A (so the form is equal to x"Ax), (ii) find matrices P, orthonormal, and D so that D diagonal, (iii) write the quadratic form in terms of y1,..., Yn where x = decide whether the form is positive/negative definite/semidefinite or indefinite. (a) 2x – 2x1x2 + 2x3 (b) –xỉ + 4x1x2 – 4x3 (c) 3x? + 4.x3 + 5xž + 4x1x2 – 4x2X3 (it may help to show that A characteristic polynomial of A). РТ АР is Py and (iv) 1 is a root of the %3D

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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1. For the following quadratic forms (i) write down the symmetric matrix A (so the form is equal to x"Ax), (ii) find matrices P, orthonormal, and D so that D = PT AP is diagonal, (iii) write the quadratic form in terms of y1,..., Yn where x = Py and (iv) decide whether the form is positive/negative definite/semidefinite or indefinite. (a) 2x – 2x1x2 + 2x3 (b) —г? + 4122 — 4л2 (c) 3xỉ + 4x3 + 5x3 + 4x1x2 – 4x2x3 (it may help to show that A = 1 is a root of the characteristic polynomial of A). -
Transcribed Image Text:1. For the following quadratic forms (i) write down the symmetric matrix A (so the form is equal to x"Ax), (ii) find matrices P, orthonormal, and D so that D = PT AP is diagonal, (iii) write the quadratic form in terms of y1,..., Yn where x = Py and (iv) decide whether the form is positive/negative definite/semidefinite or indefinite. (a) 2x – 2x1x2 + 2x3 (b) —г? + 4122 — 4л2 (c) 3xỉ + 4x3 + 5x3 + 4x1x2 – 4x2x3 (it may help to show that A = 1 is a root of the characteristic polynomial of A). -
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