6.1 What equation gives the eigenvalues of A without involving the eigenvectors ? How would you then find the eigenvectors ?
6.2 If A is singular what does this say about its eigenvalues ?
6.3 If A times A equals 4A, what numbers can be eigenvalues of A ?
6.4 Find a real matrix that has no real eigenvalues or eigenvectors.
6.5 How can you find the sum and product of the eigenvalues directly from A ?
6.6 What are the eigenvalues of the rank one matrix [1 2 1]^T [1 1 1] ?
6.7 Explain the diagonalization formula A = S LAMBDA inv(S). Why is it true and when is it true ?
6.8 What is the difference between the algebraic and geometric multiplicities of an eigenvalue of A ? Which might be larger ?
6.9 Explain why the trace of AB equals the trace of BA.
6.10 How do the eigenvectors of A help to solve du/dt = Au ?
6.11 How do the eigenvectors of A help to solve u_k+1 = Au_k ?
6.12 Define the matrix exponential e^A and its inverse and its square.
6.13 If A is symmetric, what is special about its eigenvectors ? Do any other matrices have eigenvectors with this property ?
6.14 What is the diagonalization formula when A is symmetric ?
6.15 What does it mean to say that A is "positive definite" ?
6.16 When is B = A^T A a positive definite matrix (A is real) ?
6.17 If A is positive definite describe the surface x^T A x = 1 in R^n.
6.18 What does it mean for A and B to be "similar" ? What is sure to be the same for A and B ?
6.19 The 3 by 3 matrix with ones for i>= j has what Jordan form ?
6.20 The SVD expresses A as a product of what three types of matrices ?
6.21 How is the SVD for A linked to A^T A ?
7.1 Define a linear transformation from R^3 to R^2 and give one example.
7.2 If the upper middle house on the cover of the book is the original, find something nonlinear in the transformations of the other eight houses.
7.3 If a linear transformation takes every vector in the input basis into the next basis vector (and the last into zero), what is its matrix ?
7.4 Suppose we change from the standard basis (the columns of I) to the basis given by the columns of A (invertible matrix). What is the change of basis matrix M ?
7.5 Suppose our new basis is formed from the eigenvectors of a matrix A. What matrix represents A in this new basis ?
7.6 If A and B are the matrices representing linear transformations S and T on R^n, what matrix represents the transformation from v to S(T(v)) ?
7.7 Describe five important factorizations of a matrix A and explain when each of them succeeds (what conditions on A ?).