ENGG 2013 Unit 19 The principal axes theorem

ENGG 2013 Unit 19 The principal axes theorem Mar, 2011.

Outline • Special matrices – Symmetric, skew-symmetric, orthogonal • Principle axes theorem • Application to conic sections kshum ENGG 2013 2

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 kshum ENGG 2013 3

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 because the matrix is invertible kshum by definition ENGG 2013 4

Proof continued and kshum ENGG 2013 5

Complex eigenvalue • There are some matrices whose eigenvalues are complex numbers. – For example: the matrix which represents rotation by 45 degree counter-clockwise. kshum ENGG 2013 6

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. kshum ENGG 2013 7

Matrix in special form • • Symmetric: AT=A. Skew-symmetric: AT= –A. Orthogonal: AT =A-1, or equivalently AT A = I. Examples: symmetric kshum skew-symmetric ENGG 2013 symmetric and orthogonal 8

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

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

Principal axes theorem Given any n n symmetric matrix A, we have: 1. The eigenvalues of A are real. 2. A is diagonalizable. 3. We can pick n mutually perpendicular (aka orthogonal) eigenvectors. Q Proof omitted. http: //en. wikipedia. org/wiki/Principal_axis_theorem kshum ENGG 2013 11

Two sufficient conditions for diagonalizability Symmetric, skew-symmetric, orthogonal Distinct eigenvalues Diagonalizable kshum ENGG 2013 12

Example kshum ENGG 2013 13

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 14

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

Application to conic sections • Ellipse : x 2/a + y 2/b = 1. • Hyperbola : x 2/a – y 2/b = 1. • Parabola y = ax 2. kshum ENGG 2013 16

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

Application to conic sections Diagonalize Change coordinates Hyperbola kshum 18

x 2 – 4 xy +2 y 2 = 1 kshum 19

2 x 2 + 2 xy + 2 y 2 = 1 Rewrite using symmetric matrix Compute the characteristic polynomial Find the eigenvalues kshum 20

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

2 x 2 + 2 xy + 2 y 2 = 1 v u kshum 22