Kenneth Kuttler

90 CHAPTER 5. SOME IMPORTANT LINEAR ALGEBRA

thus putting a 2×2 identity matrix in the upper left corner rather than a one. Repeating thisprocess with the above modification for the case of a complex eigenvalue leads eventuallyto 5.8.26 where Q is the product of real unitary matrices Qi above. Finally,

λ I−T =

 λ I1−P1 · · · ∗. . .

...0 λ Ir−Pr

where Ik is the 2×2 identity matrix in the case that Pk is 2×2 and is the number 1 in thecase where Pk is a 1× 1 matrix. Now, it follows that det(λ I−T ) = ∏

rk=1 det(λ Ik−Pk) .

Therefore, λ is an eigenvalue of T if and only if it is an eigenvalue of some Pk. This provesthe theorem since the eigenvalues of T are the same as those of A because they have thesame characteristic polynomial due to the similarity of A and T.

Definition 5.8.5 When a linear transformation, A, mapping a linear space, V to V hasa basis of eigenvectors, the linear transformation is called non defective. Otherwise it iscalled defective. An n×n matrix, A, is called normal if AA∗ = A∗A. An important class ofnormal matrices is that of the Hermitian or self adjoint matrices. An n×n matrix, A is selfadjoint or Hermitian if A = A∗.

The next lemma is the basis for concluding that every normal matrix is unitarily similarto a diagonal matrix.

Lemma 5.8.6 If T is upper triangular and normal, then T is a diagonal matrix.

Proof: Since T is normal, T ∗T = T T ∗. Writing this in terms of components and usingthe description of the adjoint as the transpose of the conjugate, yields the following for theikth entry of T ∗T = T T ∗.

∑j

ti jt∗jk = ∑j

ti jtk j = ∑j

t∗i jt jk = ∑j

t jit jk.

Now use the fact that T is upper triangular and let i = k = 1 to obtain the following fromthe above.

∑j

∣∣t1 j∣∣2 = ∑

∣∣t j1∣∣2 = |t11|2

You see, t j1 = 0 unless j = 1 due to the assumption that T is upper triangular. This showsT is of the form 

∗ 0 · · · 00 ∗ · · · ∗...

. . . . . ....

0 · · · 0 ∗

 .

Now do the same thing only this time take i = k = 2 and use the result just established.Thus, from the above,

∑j

∣∣t2 j∣∣2 = ∑

∣∣t j2∣∣2 = |t22|2 ,

90 CHAPTER 5. SOME IMPORTANT LINEAR ALGEBRAthus putting a 2 x 2 identity matrix in the upper left corner rather than a one. Repeating thisprocess with the above modification for the case of a complex eigenvalue leads eventuallyto 5.8.26 where Q is the product of real unitary matrices Q; above. Finally,AL _ P, tee *AI-T= . :0 Al, —P,where J; is the 2 x 2 identity matrix in the case that P; is 2 x 2 and is the number | in thecase where P; is a 1 x 1 matrix. Now, it follows that det(A7—T) = []{_, det(Ah —P;).Therefore, A is an eigenvalue of T if and only if it is an eigenvalue of some P;. This provesthe theorem since the eigenvalues of T are the same as those of A because they have thesame characteristic polynomial due to the similarity of A and 7.Definition 5.8.5 When a linear transformation, A, mapping a linear space, V to V hasa basis of eigenvectors, the linear transformation is called non defective. Otherwise it iscalled defective. Ann X n matrix, A, is called normal if AA* = A*A. An important class ofnormal matrices is that of the Hermitian or self adjoint matrices. Ann x n matrix, A is selfadjoint or Hermitian if A = A’.The next lemma is the basis for concluding that every normal matrix is unitarily similarto a diagonal matrix.Lemma 5.8.6 [fT is upper triangular and normal, then T is a diagonal matrix.Proof: Since T is normal, 7*7 = TT*. Writing this in terms of components and usingthe description of the adjoint as the transpose of the conjugate, yields the following for theik!” entry of T*T =TT*.ite = Lites = Vtiitin = Veiitie-Jj J J JjNow use the fact that T is upper triangular and let i= k = 1 to obtain the following fromthe above. 5 5Vila =L tal = la?J JYou see, tj; = 0 unless j = 1 due to the assumption that T is upper triangular. This showsT is of the forman) 0QO x *0 O xNow do the same thing only this time take i = k = 2 and use the result just established.Thus, from the above,Llei? =Leel? =lee!,J J