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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 30E

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Problem #6: Make a change of variable, x = Py, that transforms the following quadratic form into one with no cross-product
terms.
-6x² − 6x² − 6x² − 6x² − 4x1x3 + 24x1x4 +24x2x3 − 4x2x4
Construct P
For the associated quadratic form, enter the eigenvalues in decreasing order (including multiplicities) into the
answer box below, separated with commas.

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In this problem, if you give decimal answers then give at least three digits of accuracy beyond the decimal. The matrix has the following complex eigenvalues (give your answer as a comma separated list of complex numbers; use "i" for ✓-1 and feel free to use a computer to solve the relevant quadratic equation): λ = 1.65+1.548386257i, 1.65-1.548386257i Since A has non-real eigenvalues, it is not diagonalizable, but we can find a matrix C = section 5.5 of Lay): C = P = cos(6) A = sin(8) -sin(8) cos(8) [113] 1.8 Further, we may factor Cas C = XY, where X is a matrix that scales by the positive real number radians, with —π/2 ≤0 ≤ π/2) is and an invertible matrix P such that A = PCP-¹ (I want you to cook up C, P as in and Y is rotation matrix whose counter-clockwise rotation angle (in This means that if we let B be the basis for R² consisting of the columns of P, then the B-matrix of A is C. So if we're willing to change our basis for IR², then the linear transformation x Ax really is just…
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Make a change of variable, x = Py, that transforms the following quadratic form into one with no cross-product terms. 6x² + 6x² + 8x1x2 + 6x1x3 + 6x1x4 + 6x2x3 + 6x2x4 + 20x3x4 Construct P using the methods of Section 7.1. For the associated quadratic form, enter the eigenvalues in decreasing order (including multiplicities) into the answer box below, separated with commas.
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