§1.1.5 Linear Operators
Christopher Crawford
PHY 311
2014-01-17
Outline
• Linear operators
Characterization – matrices/tensors
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• 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)
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Homework #1
• Relational vs. parametric equations of curves/surfaces
• Example: circle
• Examples of physical vector spaces?
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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.)
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Change of coordinates
• Two ways of thinking about transformations
– Both yield the same rotation matrix! Difference is in what you rotate.
ACTIVE
PASSIVE
– Single basis fixed
– Actively rotate vector
– Easier to calculate matrix?
– Physical vector still (passive)
– Rotate basis vectors
– Easier to compose rotations?
EXAMPLE: 2-d rotation
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Passive transform: basis/components
• What is the meaning of `contravariant components’?
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Passive transform: basis/components
• What is the meaning of `contravariant components’?
• Example: simple rotation
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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:
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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:
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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:
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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:
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Illustration of Eigenspace
• Symmetric matrix S with eigenvectors vi , eigenvalues λi
• Similarity transform – change of basis to diagonalize S
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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
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Exponential map
• Taking little steps over and over again…
• Euler’s identity
• i is the generator of rotations
• Polar decomposition: symmetry under complex conjugation
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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’
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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
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