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## ENGG2013 Unit 19 The principal axes theorem

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**ENGG2013 Unit 19The principal axes theorem**Mar, 2011.**Outline**• Special matrices • Symmetric, skew-symmetric, orthogonal • Principle axes theorem • Application to conic sections ENGG2013**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**Theorem**An nn 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 33 case. The three columns are linearly independent becausethe matrix is invertible by definition ENGG2013**Proof continued**and and ENGG2013**Complex eigenvalue**• There are some matrices whose eigenvalues are complex numbers. • For example: the matrix which represents rotation by 45 degree counter-clockwise. ENGG2013**Theorem**If an nn matrix M has n distinct eigenvalues, then M is diagonalizable The converse is false: There is some diagonalizable matrix with repeated eigenvalues. ENGG2013**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**Orthogonal matrix**Dot product = 1 A matrix M is called orthogonal if Each column has norm 1 I MT M kshum**Orthogonal matrix**Dot product = 0 A matrix M is called orthogonal if Any two distinct columns are orthogonal kshum**Principal axes theorem**Given any nn 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**Two sufficient conditions for diagonalizability**Distinct eigenvalues Symmetric, skew-symmetric, orthogonal Diagonalizable ENGG2013**Example**ENGG2013**Similarity**Definition: We say that two nn 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**More examples**is similar to because and are similar. kshum**Application to conic sections**• Ellipse : x2/a + y2/b = 1. • Hyperbola : x2/a – y2/b = 1. • Parabola y = ax2. ENGG2013**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**Application to conic sections**Change coordinates Hyperbola Diagonalize kshum**x2 – 4xy +2y2 = 1**kshum**2x2 + 2xy + 2y2 = 1**Rewrite using symmetric matrix Compute the characteristic polynomial Find the eigenvalues kshum**2x2 + 2xy + 2y2 = 1**Columns of P are eigenvectors,normalized to norm 1. Diagonalize Change of variables kshum**2x2 + 2xy + 2y2 = 1**v u kshum