Then $S^{-1}AS = B$ where $lambda_{j}$ is the eigenvalue associated with $vec{v}_{j}$.
EXERCISES 7.3
GOAL For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.
For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology.
21. Find a 2 x 2 matrix A for which $E_{1} = ext{span}egin{bmatrix} 1 \ 2 end{bmatrix}$ and $E_{2} = ext{span}egin{bmatrix} 2 \ 3 end{bmatrix}$.
How many such matrices are there?
22. Find all 2 x 2 matrices A for which $E_{7} = mathbb{R}^{2}$.
23. Find all eigenvalues and eigenvectors of $A = egin{bmatrix} 1 & 1 \ 0 & 1 end{bmatrix}$. Is there an eigenbasis? Interpret your result geometrically.
24. Find a 2 x 2 matrix A for which $E_{1} = ext{span}egin{bmatrix} 2 \ 1 end{bmatrix}$ is the only eigenspace.
25. What can you say about the geometric multiplicity of the eigenvalues of a matrix of the form $A = egin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ a & b & c end{bmatrix}$, where a, b, c are arbitrary constants?
26. Show that if a 6 x 6 matrix A has a negative determinant, then A has at least one positive eigenvalue. Hint: Sketch the graph of the characteristic polynomial.
27. Consider a 2 x 2 matrix A. Suppose that tr A = 5 and det A = 6. Find the eigenvalues of A.
28. Consider the matrix $J_{n}(k) = egin{bmatrix} k & 1 & 0 & dots & 0 & 0 \ 0 & k & 1 & dots & 0 & 0 \ 0 & 0 & k & dots & 0 & 0 \ vdots & vdots & vdots & ddots & vdots & vdots \ 0 & 0 & 0 & dots & k & 1 \ 0 & 0 & 0 & dots & 0 & k end{bmatrix}$ (with all k's on the diagonal and 1's directly above), where k is an arbitrary constant. Find the eigenvalue(s) of $J_{n}(k)$, and determine their algebraic and geometric multiplicities.