1. §1.1. 5 Linear Operators Christopher Crawford PHY 311 2014-01-17

2. Outline Linear operators Characterization – matrices/tensors Change of basis – coordinate transform of vectors Active vs. passive rotations Basis vs. components – `contravariant’ components Properties of orthogonal transformations Eigenparaphernalia Eigen-{vectors, values, spaces} Similarity transform – coordinate transform of matrices Properties of Hermitian (symmetric) operators Polar decomposition – put it all together! Exponential map, symmetry Singular value decomposition (SVD) 2

3. Homework #1 Relational vs. parametric equations of curves/surfaces Example: circle Examples of physical vector spaces? 3

4. Linear transformation A function which preserves linear combinations – What does that buy us? – Is the real world linear? – Relation to dot and cross products and to determinant? Tensor structure – matrix and two sets of bases – Rows of matrix = image of basis vectors – Determinant = expansion volume (triple prod.) 4

5. Change of coordinates Two ways of thinking about transformations – Both yield the same rotation matrix! Difference is in what you rotate. 5 ACTIVE – Single basis fixed – Actively rotate vector – Easier to calculate matrix? PASSIVE – Physical vector still (passive) – Rotate basis vectors – Easier to compose rotations? EXAMPLE: 2-d rotation

6. Passive transform: basis/components What is the meaning of `contravariant components’? 6

7. Passive transform: basis/components What is the meaning of `contravariant components’? Example: simple rotation 7

8. Orthogonal transformations R is orthogonal (rotation) if it `preserves the metric’ (distances and angles are the same before and after Development in terms of basis: 8

9. Orthogonal transformations R is orthogonal (rotation) if it `preserves the metric’ (distances and angles are the same before and after Development in terms of basis: 9

10. Orthogonal transformations R is orthogonal (rotation) if it `preserves the metric’ (distances and angles are the same before and after Development in terms of basis: Development in terms of components: 10

11. Orthogonal transformations R is orthogonal (rotation) if it `preserves the metric’ (distances and angles are the same before and after Development in terms of basis: Development in terms of components: Starting with an orthonormal basis: 11

12. Illustration of Eigenspace Symmetric matrix S with eigenvectors v i, eigenvalues λ i Similarity transform – change of basis to diagonalize S 12

13. Properties of symmetric matrices A symmetric (Hermitian) matrix has real eigenvalues – What about an antisymmetric / orthogonal matrix? Eigenvectors of a symmetric (Hermitian) matrix with distinct eigenvalues are orthogonal 13

14. Exponential map Taking little steps over and over again… Euler’s identity i is the generator of rotations Polar decomposition: symmetry under complex conjugation 14

15. Polar decomposition Apply the same principle from complex numbers to operators Not quite so clean due to noncommutativity of matrices This is the basic example of a `Lie algebra’ 15

16. Singular value decomposition (SVD) Breaks down the structure of any m x n matrix into rotations, a stretch and a projection Extremely useful in numerical routines 16