CS223 · Chapter 6 · Annotated SlidesOrthogonal Sets and Orthogonal Matrices
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Orthogonal Sets and Orthogonal Matrices

Orthogonal and orthonormal sets and bases, weights in an orthogonal basis cⱼ = (y · uⱼ)/(uⱼ · uⱼ), and matrices with orthonormal columns (UᵀU = I).

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All 6 slides
  1. 1 A set of vectors is an orthonormal set if all the following are true:
  2. 2 Slide 2
  3. 3 Slide 3
  4. 4 Orthogonal Matrix
  5. 5 Slide 5
  6. 6 Slide 6
Slide 1 / 6

A set of vectors is an orthonormal set if all the following are true:

Slide 1: A set of vectors is an orthonormal set if all the following are true:

Notes

  1. An orthogonal set of nonzero vectors is linearly independent. In an orthogonal basis, y = Σ cⱼuⱼ with cⱼ = (y · uⱼ)/(uⱼ · uⱼ): no row reduction needed.

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Slide 2

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Slide 3

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Slide 4 / 6

Orthogonal Matrix

Slide 4: Orthogonal Matrix

Notes

  1. Orthonormal columns ⇔ UᵀU = I. A square such U is an orthogonal matrix: U⁻¹ = Uᵀ, and it preserves lengths and angles.

Exam angle

PredictionQ1

For an orthogonal (square) matrix U, U⁻¹ equals:

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Slide 6

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