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

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)

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

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.)

5. Change of coordinates Two ways of thinking about transformations
Both yield the same rotation matrix! Difference is in what you rotate.
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’?

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

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:

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:

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:

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:

12. Illustration of Eigenspace
Symmetric matrix S with eigenvectors vi , eigenvalues λi
Similarity transform – change of basis to diagonalize S

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

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

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’

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