C H A P T E R 8 Canonical Forms Recall that at the beginning of Section 7.5 we stated that a canonical form for T ∞ L(V) is simply a representation in which the matrix takes on an especially simple form. For example, if there exists a basis of eigenvectors of T, then the matrix representation will be diagonal. In this case, it is then quite trivial to assess the various properties of T such as its rank, determinant and eigenval- ues. Unfortunately, while this is generally the most desirable form for a matrix representation, it is also generally impossible to achieve. We now wish to determine certain properties of T that will allow us to learn as much as we can about the possible forms its matrix representation can take. There are three major canonical forms that we will consider in this chapter : triangular, rational and Jordan. (This does not count the Smith form, which is really a tool, used to find the rational and Jordan forms.) As we have done before, our approach will be to study each of these forms in more than one way. By so doing, we shall gain much insight into their meaning, as well as learning additional techniques that are of great use in various branches of mathematics. 8.1 ELEMENTARY CANONICAL FORMS In order to ease into the subject, this section presents a simple and direct method of treating two important results: the triangular form for complex matrices and the diagonalization of normal matrices. To begin with, suppose 382 8.1 ELEMENTARY CANONICAL FORMS 383 that we have a matrix A ∞ Mñ(ç). We define the adjoint (or Hermitian adjoint) of A to be the matrix A¿ = A*T. In other words, the adjoint of A is its complex conjugate transpose. From Theorem 3.18(d), it is easy to see that (AB)¿ = B¿A¿ . If it so happens that A¿ = A, then A is said to be a Hermitian matrix. If a matrix U ∞ Mñ(ç) has the property that U¿ = Uî, then we say that U is unitary. Thus a matrix U is unitary if UU¿ = U¿U = I. (Note that by Theorem 3.21, it is only necessary to require either UU¿ = I or U¿U = I.) We also see that the product of two unitary matrices U and V is unitary since (UV)¿UV = V¿U¿UV = V¿IV = V¿V = I. If a matrix N ∞ Mñ(ç) has the property that it commutes with its adjoint, i.e., NN¿ = N¿N, then N is said to be a normal matrix. Note that Hermitian and unitary matrices are auto- matically normal. Example 8.1 Consider the matrix A ∞ Mì(ç) given by 1 "1 !1% A = $ '!!. 2 #i !!i& Then the adjoint of A is given by † 1 "!!1 !i% A = $ ' 2 #!1 !i& and we leave it to the reader to verify that AA¿ = A¿A = 1, and hence show that A is unitary. ∆ We will devote considerable time in Chapter 10 to the study of these matrices. However, for our present purposes, we wish to point out one impor- tant property of unitary matrices. Note that since U ∞ Mñ(ç), the rows Uá and columns Ui of U are just vectors in çn. This means that we can take their inner product relative to the standard inner product on çn (see Example 2.9). Writing out the relation UU¿ = I in terms of components, we have n n n † † (UU )ij = !uiku kj = !uiku jk* = !u jk*uik = U j ,!Ui = "ij k=1 k=1 k=1 and from U¿U = I we see that n n † † i j (U U)ij = !u ikukj = !uki*ukj = U ,!U = "ij !!. k=1 k=1 384 CANONICAL FORMS In other words, a matrix is unitary if and only if its rows (or columns) each form an orthonormal set. Note we have shown that if the rows (columns) of U ∞ Mñ(ç) form an orthonormal set, then so do the columns (rows), and either of these is a sufficient condition for U to be unitary. For example, the reader can easily verify that the matrix A in Example 8.1 satisfies these conditions. It is also worth pointing out that Hermitian and unitary matrices have important analogues over the real number system. If A ∞ Mñ(®) is Hermitian, then A = A¿ = AT, and we say that A is symmetric. If U ∞ Mñ(®) is unitary, then Uî = U¿ = UT, and we say that U is orthogonal. Repeating the above calculations over ®, it is easy to see that a real matrix is orthogonal if and only if its rows (or columns) form an orthonormal set. It will also be useful to recall from Section 3.6 that if A and B are two matrices for which the product AB is defined, then the ith row of AB is given by (AB)á = AáB and the ith column of AB is given by (AB)i = ABi. We now prove yet another version of the triangular form theorem. Theorem 8.1 (Schur Canonical Form) If A ∞ Mñ(ç), then there exists a unitary matrix U ∞ Mñ(ç) such that U¿AU is upper-triangular. Furthermore, the diagonal entries of U¿AU are just the eigenvalues of A. Proof If n = 1 there is nothing to prove, so we assume that the theorem holds for any square matrix of size n - 1 ˘ 1, and suppose A is of size n. Since we are dealing with the algebraically closed field ç, we know that A has n (not necessarily distinct) eigenvalues (see Section 7.3). Let ¬ be one of these eigenvalues, and denote the corresponding eigenvector by Uÿ1. By Theorem 2.10 we extend Uÿ1 to a basis for çn, and by the Gram-Schmidt process (Theorem 2.21) we assume that this basis is orthonormal. From our discussion above, we see that this basis may be used as the columns of a unitary matrix Uÿ with Uÿ1 as its first column. We then see that (U! †AU! )1 = U! †(AU! )1 = U! †(AU! 1) = U! †(!U! 1) = !(U! †U! 1) ! † ! 1 1 = !(U U) = !I and hence Uÿ¿AUÿ has the form " ! *!!"!!* % $ ' $ 0 ! ! ' U! †AU! = $ # ! B ! ' $ ' $ 0 !! !! ' # & 8.1 ELEMENTARY CANONICAL FORMS 385 where B ∞ Mn-1(ç) and the *’s are (in general) nonzero scalars. By our induc- tion hypothesis, we may choose a unitary matrix W ∞ Mn-1(ç) such that W¿BW is upper-triangular. Let V ∞ Mñ(ç) be a unitary matrix of the form ! 1 0!!!!!0 $ # & # 0 ! ! & V = # " ! W ! & # & # 0 !! !! & " % and define the unitary matrix U = UÿV ∞ Mñ(ç). Then U¿AU = (UÿV)¿A(UÿV) = V¿(Uÿ¿AUÿ)V is upper-triangular since (in an obvious shorthand notation) ! 1 0 $!' *$!1 0 $ ! 1 0 $!' * $ V †(U! †AU! )V = # † &# &# & = # † &# & " 0 W %"0 B%"0 W % " 0 W %"0 BW % !' * $ = # & 0 W †BW " % and W¿BW is upper-triangular by the induction hypothesis. Noting that ¬I - U¿AU is upper-triangular, it is easy to see (using Theorem 4.5) that the roots of det(¬I - U¿AU) are just the diagonal entries of U¿AU. But det(¬I - U¿AU) = det[U¿(¬I - A)U] = det(¬I - A) so that A and U¿AU have the same eigenvalues. ˙ Corollary If A ∞ Mñ(®) has all its eigenvalues in ®, then the matrix U defined in Theorem 8.1 may be chosen to have all real entries. Proof If ¬ ∞ ® is an eigenvalue of A, then A - ¬I is a real matrix with deter- minant det(A - ¬I) = 0, and therefore the homogeneous system of equations (A - ¬I)X = 0 has a real solution. Defining Uÿ1 = X, we may now proceed as in Theorem 8.1. The details are left to the reader (see Exercise 8.8.1). ˙ We say that two matrices A, B ∞ Mñ(ç) are unitarily similar (written A – B) if there exists a unitary matrix U such that B = U¿AU = UîAU. Since this 386 CANONICAL FORMS defines an equivalence relation on the set of all matrices in Mñ(ç), many authors say that A and B are unitarily equivalent. However, we will be using the term “equivalent” in a somewhat more general context later in this chapter, and the word “similar” is in accord with our earlier terminology. We leave it to the reader to show that if A and B are unitarily similar and A is normal, then B is also normal (see Exercise 8.8.2). In particular, suppose that U is unitary and N is such that U¿NU = D is diagonal. Since any diagonal matrix is automatically normal, it follows that N must be normal also. We now show that the converse is also true, i.e., that any normal matrix is unitarily similar to a diagonal matrix. To see this, suppose N is normal, and let U¿NU = D be the Schur canoni- cal form of N. Then D is both upper-triangular and normal (since it is unitarily similar to a normal matrix). We claim that the only such matrices are diagonal. For, consider the (1, 1) elements of DD¿ and D¿D. From what we showed above, we have (DD¿)èè = ÓDè, DèÔ = \dèè\2 + \dèì\2 + ~ ~ ~ + \dèñ\2 and (D¿D)èè = ÓD1, D1Ô = \dèè\2 + \dìè\2 + ~ ~ ~ + \dñè\2 . But D is upper-triangular so that dìè = ~ ~ ~ = dñè = 0. By normality we must have (DD¿)èè = (D¿D)èè, and therefore dèì = ~ ~ ~ = dèñ = 0 also. In other words, with the possible exception of the (1, 1) entry, all entries in the first row and column of D must be zero.