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ENGG2013 Unit 19 The principal axes theorem

ENGG2013 Unit 19 The principal axes theorem

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ENGG2013 Unit 19 The principal axes theorem

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  1. ENGG2013 Unit 19The principal axes theorem Mar, 2011.

  2. Outline • Special matrices • Symmetric, skew-symmetric, orthogonal • Principle axes theorem • Application to conic sections ENGG2013

  3. Diagonalizable ?? • A square matrix M is called diagonalizable if we can find an invertible matrix, say P, such that the product P–1 M P is a diagonal matrix. • Example • Some matrix cannot be diagonalized. • Example ENGG2013

  4. Theorem An nn matrix M is diagonalizable if and only if we can find n linear independent eigenvectors of M. Proof: For concreteness, let’s just consider the 33 case. The three columns are linearly independent becausethe matrix is invertible by definition ENGG2013

  5. Proof continued and and ENGG2013

  6. Complex eigenvalue • There are some matrices whose eigenvalues are complex numbers. • For example: the matrix which represents rotation by 45 degree counter-clockwise. ENGG2013

  7. Theorem If an nn matrix M has n distinct eigenvalues, then M is diagonalizable The converse is false: There is some diagonalizable matrix with repeated eigenvalues. ENGG2013

  8. Matrix in special form • Symmetric: AT=A. • Skew-symmetric: AT= –A. • Orthogonal: AT =A-1, or equivalently AT A= I. • Examples: symmetric and orthogonal symmetric skew-symmetric ENGG2013

  9. Orthogonal matrix Dot product = 1 A matrix M is called orthogonal if Each column has norm 1 I MT M kshum

  10. Orthogonal matrix Dot product = 0 A matrix M is called orthogonal if Any two distinct columns are orthogonal kshum

  11. Principal axes theorem Given any nn symmetric matrix A, we have: • The eigenvalues of A are real. • A is diagonalizable. • We can pick n mutually perpendicular (aka orthogonal) eigenvectors.  Q Proof omitted. http://en.wikipedia.org/wiki/Principal_axis_theorem ENGG2013

  12. Two sufficient conditions for diagonalizability Distinct eigenvalues Symmetric, skew-symmetric, orthogonal Diagonalizable ENGG2013

  13. Example ENGG2013

  14. Similarity Definition: We say that two nn matrix A and B are similar if we can find an invertible matrix S such that Example: and are similar, The notion of diagonalization can be phrased in terms of similarity: matrix A is diagonalizable if and only if A is similar to a diagonal matrix. kshum

  15. More examples is similar to because and are similar. kshum

  16. Application to conic sections • Ellipse : x2/a + y2/b = 1. • Hyperbola : x2/a – y2/b = 1. • Parabola y = ax2. ENGG2013

  17. Application to conic sections Is x2 – 4xy +2y2 = 1 a ellipse, or a hyperbola? Rewrite using symmetric matrix Find the characteristic polynomial Solve for the eigenvalues kshum

  18. Application to conic sections Change coordinates Hyperbola Diagonalize kshum

  19. x2 – 4xy +2y2 = 1 kshum

  20. 2x2 + 2xy + 2y2 = 1 Rewrite using symmetric matrix Compute the characteristic polynomial Find the eigenvalues kshum

  21. 2x2 + 2xy + 2y2 = 1 Columns of P are eigenvectors,normalized to norm 1. Diagonalize Change of variables kshum

  22. 2x2 + 2xy + 2y2 = 1 v u kshum