Go to CCP Homepage Go to Materials Page Go to Linear Algebra Materials Go to Table of Contents
Go Back One Page

Orthogonality

Part 5: Summary

  1. If an m x n matrix A (not necessarily square) has orthonormal columns, what can you say about ATA?

  2. If A is an orthogonal matrix, what is its inverse?

  3. Suppose W is a subspace of Rn, and you know an orthogonal basis for W. If y is a vector not in W, how do you compute the projection of y on W? How do you find the component of y in Wperp? How do you find the distance from y to W? To what extent are these questions easier to answer if you know an orthonormal basis for W?

  4. If A is an n x n orthogonal matrix, and x and y are n-vectors, how is the dot product of x and y related to the dot product of Ax and Ay? How is the length of x related to the length of Ax? How is the second of these questions related to the first?

  5. What numbers can be eigenvalues of an orthogonal matrix? What numbers can be determinants? How is the second of these questions related to the first?
Go to CCP Homepage Go to Materials Page Go to Linear Algebra Materials Go to Table of Contents
Go Back One Page

modules at math.duke.edu