Lecture 3: Orthonormal Columns in Q Give Q’Q = I
Description
This lecture focuses on orthogonal matrices and subspaces. Professor Strang reviews the four fundamental subspaces: column space C(A), row space C(AT), nullspace N(A), left nullspace N(AT).
Summary
Examples:
- Rotations
- Reflections
- Hadamard matrices
- Haar wavelets
- Discrete Fourier Transform (DFT)
- Complex inner product
Related section in textbook: I.5
Instructor: Prof. Gilbert Strang
Problems for Lecture 3
From textbook Section I.5
2. Draw unit vectors u and v that are not orthogonal. Show that w = v - u(uTv) is orthogonal to u (and add w to your picture).
4. Key property of every orthogonal matrix: ||Qx||2 = ||x||2 for every vector x. More than this, show that (Qx)T(Qy)= xTy for every vector x and y. So lengths and angles are not changed by Q. Computations with Q never overflow!
6. A permutation matrix has the same columns as the identity matrix (in some order). Explain why this permutation matrix and every permutation matrix is orthogonal :
P = \(\left[\begin{array}{@{\,}rrrr@{\,}}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\\\end{array}\right]\) has orthonormal columns so P_T_P = ___ and _P_−1= ___.
When a matrix is symmetric or orthogonal, it will have orthogonal eigenvectors. This is the most important source of orthogonal vectors in applied mathematics.
