Linear Algebra

Section II. Similarity 353(b) Is the previous item a coincidence? Or can we always switch the P and theP −1 ?2.13 Show that the P used to diagonalize in Example 2.5 is not unique.2.14 Find a formula for the powers of this matrix Hint: see Exercise 8.( )−3 1−4 2̌ 2.15 Diagonalize ( ) these.1 1(a)(b)0 0( )0 11 02.16 We can ask how diagonalization interacts with the matrix operations. Assumethat t, s: V → V are each diagonalizable. Is ct diagonalizable for all scalars c?What about t + s? t ◦ s?̌ 2.17 Show that matrices of this form are not diagonalizable.( )1 cc ≠ 00 12.18 Show ( that ) each of these ( is)diagonalizable.1 2x y(a)(b)x, y, z scalars2 1y zII.3 Eigenvalues and EigenvectorsIn this subsection we will focus on the property of Corollary 2.4.3.1 Definition A transformation t: V → V has a scalar eigenvalue λ if thereis a nonzero eigenvector ⃗ ζ ∈ V such that t( ⃗ ζ) = λ · ⃗ζ.(“Eigen” is German for “characteristic of” or “peculiar to”; some authors callthese characteristic values and vectors. No authors call them “peculiar”.)3.2 Example The projection map⎛ ⎞ ⎛ ⎞⎝ x yz⎠π↦−→⎝ x y0⎠x, y, z ∈ Chas an eigenvalue of 1 associated with any eigenvector of the form⎛ ⎞x⎝y⎠0where x and y are non-0 scalars. On the other hand, 2 is not an eigenvalue ofπ since no non-⃗0 vector is doubled.That example shows why the ‘non-⃗0’ appears in the definition. Disallowing⃗0 as an eigenvector eliminates trivial eigenvalues.