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[1 2 8. Let A = and define Q : R² → R by Q(x) = x"Ax. 2 1 (a) Find the characteristic polynomial of A. (b) Find the eigenvalues of A. (c) Find a basis for each eigenspace of A. (d) For each eigenvalue, state its algebraic and geometric multiplicity. (e) Is A orthogonally diagonalizable? If so, find a matrix P which orthogonally diagonalizes A. (f) Is the quadratic form Q positive definite, negative definite, or neither? (g) Is the conic section defined by the equation Q(x) = 1 an ellipse or an hyperbola? Why?

[1 2 8. Let A = and define Q : R² → R by Q(x) = x"Ax. 2 1 (a) Find the characteristic polynomial of A. (b) Find the eigenvalues of A. (c) Find a basis for each eigenspace of A. (d) For each eigenvalue, state its algebraic and geometric multiplicity. (e) Is A orthogonally diagonalizable? If so, find a matrix P which orthogonally diagonalizes A. (f) Is the quadratic form Q positive definite, negative definite, or neither? (g) Is the conic section defined by the equation Q(x) = 1 an ellipse or an hyperbola? Why?

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 2 8. Let A = and define Q : R² → R by Q(x) = x"Ax. 2 1 (a) Find the characteristic polynomial of A. (b) Find the eigenvalues of A. (c) Find a basis for each eigenspace of A. (d) For each eigenvalue, state its algebraic and geometric multiplicity. (e) Is A orthogonally diagonalizable? If so, find a matrix P which orthogonally diagonalizes A. (f) Is the quadratic form Q positive definite, negative definite, or neither? (g) Is the conic section defined by the equation Q(x) = 1 an ellipse or an hyperbola? Why?
Transcribed Image Text:[1 2 8. Let A = and define Q : R² → R by Q(x) = x"Ax. 2 1 (a) Find the characteristic polynomial of A. (b) Find the eigenvalues of A. (c) Find a basis for each eigenspace of A. (d) For each eigenvalue, state its algebraic and geometric multiplicity. (e) Is A orthogonally diagonalizable? If so, find a matrix P which orthogonally diagonalizes A. (f) Is the quadratic form Q positive definite, negative definite, or neither? (g) Is the conic section defined by the equation Q(x) = 1 an ellipse or an hyperbola? Why?
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