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8.7. Complex Matrices 461
8.7 Complex Matrices
If A is an n × n matrix, the characteristic polynomial cA(x) is a polynomial of degree n and the eigenvalues of A are just the roots of cA(x). In most of our examples these roots have been real numbers (in fact, the examples have been carefully chosen so this will be the case!); but it need not happen, even when
the characteristic polynomial has real coefficients. For example, if A =
[
]
then cA(x) = x^2 + 1
has roots i and −i, where i is a complex number satisfying i^2 = −1. Therefore, we have to deal with the possibility that the eigenvalues of a (real) square matrix might be complex numbers.
In fact, nearly everything in this book would remain true if the phrase real number were replaced by complex number wherever it occurs. Then we would deal with matrices with complex entries, systems of linear equations with complex coefficients (and complex solutions), determinants of complex matrices, and vector spaces with scalar multiplication by any complex number allowed. Moreover, the proofs of most theorems about (the real version of) these concepts extend easily to the complex case. It is not our intention here to give a full treatment of complex linear algebra. However, we will carry the theory far enough to give another proof that the eigenvalues of a real symmetric matrix A are real (Theorem 5.5.7) and to prove the spectral theorem, an extension of the principal axes theorem (Theorem 8.2.2).
The set of complex numbers is denoted C. We will use only the most basic properties of these numbers (mainly conjugation and absolute values), and the reader can find this material in Appendix A.
If n ≥ 1, we denote the set of all n-tuples of complex numbers by Cn. As with Rn, these n-tuples will be written either as row or column matrices and will be referred to as vectors. We define vector operations on Cn^ as follows:
(v 1 , v 2 ,... , vn) + (w 1 , w 2 ,... , wn) = (v 1 + w 1 , v 2 + w 2 ,... , vn + wn) u(v 1 , v 2 ,... , vn) = (uv 1 , uv 2 ,... , uvn) for u in C
With these definitions, Cn^ satisfies the axioms for a vector space (with complex scalars) given in Chapter 6. Thus we can speak of spanning sets for Cn, of linearly independent subsets, and of bases. In all cases, the definitions are identical to the real case, except that the scalars are allowed to be complex numbers. In particular, the standard basis of Rn^ remains a basis of Cn, called the standard basis of Cn.
A matrix A =
[
ai j
]
is called a complex matrix if every entry ai j is a complex number. The notion of conjugation for complex numbers extends to matrices as follows: Define the conjugate of A =
[
ai j
]
to be the matrix A =
[
ai j
]
obtained from A by conjugating every entry. Then (using Appendix A)
A + B = A + B and AB = A B
holds for all (complex) matrices of appropriate size.
462 Orthogonality
The Standard Inner Product
There is a natural generalization to Cn^ of the dot product in Rn.
Definition 8.15 Standard Inner Product in Rn Given z = (z 1 , z 2 ,... , zn) and w = (w 1 , w 2 ,... , wn) in Cn , define their standard inner product 〈 z, w〉 by 〈 z, w〉 = z 1 w 1 + z 2 w 2 + · · · + znwn = z · w where w is the conjugate of the complex number w.
Clearly, if z and w actually lie in Rn, then 〈z, w〉 = z · w is the usual dot product.
Example 8.7. If z = (2, 1 − i, 2i, 3 − i) and w = ( 1 − i, −1, −i, 3 + 2 i), then
〈z, w〉 = 2 ( 1 + i) + ( 1 − i)(− 1 ) + ( 2 i)(i) + ( 3 − i)( 3 − 2 i) = 6 − 6 i 〈z, z〉 = 2 · 2 + ( 1 − i)( 1 + i) + ( 2 i)(− 2 i) + ( 3 − i)( 3 + i) = 20
Note that 〈z, w〉 is a complex number in general. However, if w = z = (z 1 , z 2 ,... , zn), the definition gives 〈z, z〉 = |z 1 |^2 + · · · + |zn|^2 which is a nonnegative real number, equal to 0 if and only if z = 0. This explains the conjugation in the definition of 〈z, w〉, and it gives (4) of the following theorem.
Theorem 8.7. Let z, z 1 , w, and w 1 denote vectors in Cn , and let λ denote a complex number.
- 〈 z + z 1 , w〉 = 〈 z, w〉 + 〈 z 1 , w〉 and 〈 z, w + w 1 〉 = 〈 z, w〉 + 〈 z, w 1 〉.
- 〈 λ z, w〉 = λ 〈 z, w〉 and 〈 z, λ w〉 = λ 〈 z, w〉.
- 〈 z, w〉 = 〈 w, z〉.
- 〈 z, z〉 ≥ 0 , and 〈 z, z〉 = 0 if and only if z = 0.
Proof. We leave (1) and (2) to the reader (Exercise 8.7.10), and (4) has already been proved. To prove (3), write z = (z 1 , z 2 ,.. ., zn) and w = (w 1 , w 2 ,... , wn). Then
〈w, z〉 = (w 1 z 1 + · · · + wnzn) = w 1 z 1 + · · · + wnzn = z 1 w 1 + · · · + znwn = 〈z, w〉
464 Orthogonality
The following properties of AH^ follow easily from the rules for transposition of real matrices and extend these rules to complex matrices. Note the conjugate in property (3).
Theorem 8.7. Let A and B denote complex matrices, and let λ be a complex number.
- (AH^ )H^ = A.
- (A + B)H^ = AH^ + BH.
- ( λ A)H^ = λ AH.
- (AB)H^ = BH^ AH.
Hermitian and Unitary Matrices
If A is a real symmetric matrix, it is clear that AH^ = A. The complex matrices that satisfy this condition turn out to be the most natural generalization of the real symmetric matrices:
Definition 8.18 Hermitian Matrices A square complex matrix A is called hermitian^15 if AH^ = A , equivalently if A = AT.
Hermitian matrices are easy to recognize because the entries on the main diagonal must be real, and the “reflection” of each nondiagonal entry in the main diagonal must be the conjugate of that entry.
Example 8.7.
3 i 2 + i −i − 2 − 7 2 − i − 7 1
(^) is hermitian, whereas
[
1 i i − 2
]
and
[
1 i −i i
]
are not.
The following Theorem extends Theorem 8.2.3, and gives a very useful characterization of hermitian matrices in terms of the standard inner product in Cn.
Theorem 8.7. An n × n complex matrix A is hermitian if and only if 〈A z, w〉 = 〈 z, A w〉
for all n -tuples z and w in Cn.
(^15) The name hermitian honours Charles Hermite (1822–1901), a French mathematician who worked primarily in analysis and is remembered as the first to show that the number e from calculus is transcendental—that is, e is not a root of any polynomial with integer coefficients.
8.7. Complex Matrices 465
Proof. If A is hermitian, we have AT^ = A. If z and w are columns in Cn, then 〈z, w〉 = zT^ w, so
〈Az, w〉 = (Az)T^ w = zT^ AT^ w = zT^ Aw = zT^ (Aw) = 〈z, Aw〉
To prove the converse, let e (^) j denote column j of the identity matrix. If A =
[
ai j
]
, the condition gives ai j = 〈ei, Ae (^) j〉 = 〈Aei, e (^) j〉 = ai j
Hence A = AT^ , so A is hermitian.
Let A be an n ×n complex matrix. As in the real case, a complex number λ is called an eigenvalue of A if Ax = λ x holds for some column x 6 = 0 in Cn. In this case x is called an eigenvector of A corresponding to λ. The characteristic polynomial cA(x) is defined by
cA(x) = det (xI − A)
This polynomial has complex coefficients (possibly nonreal). However, the proof of Theorem 3.3.2 goes through to show that the eigenvalues of A are the roots (possibly complex) of cA(x).
It is at this point that the advantage of working with complex numbers becomes apparent. The real numbers are incomplete in the sense that the characteristic polynomial of a real matrix may fail to have all its roots real. However, this difficulty does not occur for the complex numbers. The so-called funda- mental theorem of algebra ensures that every polynomial of positive degree with complex coefficients has a complex root. Hence every square complex matrix A has a (complex) eigenvalue. Indeed (Appendix A), cA(x) factors completely as follows:
cA(x) = (x − λ 1 )(x − λ 2 ) · · · (x − λn)
where λ 1 , λ 2 ,... , λn are the eigenvalues of A (with possible repetitions due to multiple roots).
The next result shows that, for hermitian matrices, the eigenvalues are actually real. Because symmet- ric real matrices are hermitian, this re-proves Theorem 5.5.7. It also extends Theorem 8.2.4, which asserts that eigenvectors of a symmetric real matrix corresponding to distinct eigenvalues are actually orthogonal. In the complex context, two n-tuples z and w in Cn^ are said to be orthogonal if 〈z, w〉 = 0.
Theorem 8.7. Let A denote a hermitian matrix.
- The eigenvalues of A are real.
- Eigenvectors of A corresponding to distinct eigenvalues are orthogonal.
Proof. Let λ and μ be eigenvalues of A with (nonzero) eigenvectors z and w. Then Az = λ z and Aw = μw, so Theorem 8.7.4 gives
λ 〈z, w〉 = 〈 λ z, w〉 = 〈Az, w〉 = 〈z, Aw〉 = 〈z, μw〉 = μ〈z, w〉 (8.6)
If μ = λ and w = z, this becomes λ 〈z, z〉 = λ 〈z, z〉. Because 〈z, z〉 = ‖z‖^2 6 = 0, this implies λ = λ. Thus λ is real, proving (1). Similarly, μ is real, so equation (8.6) gives λ 〈z, w〉 = μ〈z, w〉. If λ 6 = μ, this implies 〈z, w〉 = 0, proving (2).
The principal axes theorem (Theorem 8.2.2) asserts that every real symmetric matrix A is orthogonally diagonalizable—that is PT^ AP is diagonal where P is an orthogonal matrix (P−^1 = PT^ ). The next theorem identifies the complex analogs of these orthogonal real matrices.
8.7. Complex Matrices 467
Hence the eigenvalues are 2 and 8 (both real as expected), and corresponding eigenvectors are[ 2 + i − 1
]
and
[
2 − i
]
(orthogonal as expected). Each has length
6 so, as in the (real)
diagonalization algorithm, let U = √^16
[
2 + i 1 − 1 2 − i
]
be the unitary matrix with the normalized eigenvectors as columns. Then UH^ AU =
[
]
is diagonal.
Unitary Diagonalization
An n × n complex matrix A is called unitarily diagonalizable if UH^ AU is diagonal for some unitary matrix U. As Example 8.7.6 suggests, we are going to prove that every hermitian matrix is unitarily diagonalizable. However, with only a little extra effort, we can get a very important theorem that has this result as an easy consequence.
A complex matrix is called upper triangular if every entry below the main diagonal is zero. We owe the following theorem to Issai Schur.^16
Theorem 8.7.7: Schur’s Theorem If A is any n × n complex matrix, there exists a unitary matrix U such that
UH^ AU = T
is upper triangular. Moreover, the entries on the main diagonal of T are the eigenvalues λ 1 , λ 2 ,... , λn of A (including multiplicities).
Proof. We use induction on n. If n = 1, A is already upper triangular. If n > 1, assume the theorem is valid for (n − 1 ) × (n − 1 ) complex matrices. Let λ 1 be an eigenvalue of A, and let y 1 be an eigenvector with ‖y 1 ‖ = 1. Then y 1 is part of a basis of Cn^ (by the analog of Theorem 6.4.1), so the (complex analog of the) Gram-Schmidt process provides y 2 ,... , yn such that {y 1 , y 2 ,... , yn} is an orthonormal basis of Cn. If U 1 =
[
y 1 y 2 · · · yn
]
is the matrix with these vectors as its columns, then (see Lemma 5.4.3)
U 1 H AU 1 =
[
λ 1 X 1 0 A 1
]
in block form. Now apply induction to find a unitary (n − 1 ) × (n − 1 ) matrix W 1 such that W 1 H A 1 W 1 = T 1
is upper triangular. Then U 2 =
[
0 W 1
]
is a unitary n × n matrix. Hence U = U 1 U 2 is unitary (using
Theorem 8.7.6), and
UH^ AU = U 2 H (U 1 H AU 1 )U 2 (^16) Issai Schur (1875–1941) was a German mathematician who did fundamental work in the theory of representations of
groups as matrices.
468 Orthogonality
[
0 W 1 H
] [
λ 1 X 1 0 A 1
] [
0 W 1
]
[
λ 1 X 1 W 1 0 T 1
]
is upper triangular. Finally, A and UH^ AU = T have the same eigenvalues by (the complex version of) Theorem 5.5.1, and they are the diagonal entries of T because T is upper triangular.
The fact that similar matrices have the same traces and determinants gives the following consequence of Schur’s theorem.
Corollary 8.7. Let A be an n × n complex matrix, and let λ 1 , λ 2 ,... , λn denote the eigenvalues of A , including multiplicities. Then
det A = λ 1 λ 2 · · · λn and tr A = λ 1 + λ 2 + · · · + λn
Schur’s theorem asserts that every complex matrix can be “unitarily triangularized.” However, we
cannot substitute “unitarily diagonalized” here. In fact, if A =
[
]
, there is no invertible complex
matrix U at all such that U−^1 AU is diagonal. However, the situation is much better for hermitian matrices.
Theorem 8.7.8: Spectral Theorem
If A is hermitian, there is a unitary matrix U such that UH^ AU is diagonal.
Proof. By Schur’s theorem, let UH^ AU = T be upper triangular where U is unitary. Since A is hermitian, this gives T H^ = (UH^ AU)H^ = UH^ AHUHH^ = UH^ AU = T
This means that T is both upper and lower triangular. Hence T is actually diagonal.
The principal axes theorem asserts that a real matrix A is symmetric if and only if it is orthogonally diagonalizable (that is, PT^ AP is diagonal for some real orthogonal matrix P). Theorem 8.7.8 is the complex analog of half of this result. However, the converse is false for complex matrices: There exist unitarily diagonalizable matrices that are not hermitian.
Example 8.7.
Show that the non-hermitian matrix A =
[
]
is unitarily diagonalizable.
Solution. The characteristic polynomial is cA(x) = x^2 + 1. Hence the eigenvalues are i and −i, and
it is easy to verify that
[
i − 1
]
and
[
i
]
are corresponding eigenvectors. Moreover, these
eigenvectors are orthogonal and both have length
2, so U = √^12
[
i − 1 − 1 i
]
is a unitary matrix
470 Orthogonality
Thus cA(A) = (A − λ 1 I)(A − λ 2 I)(A − λ 3 I) · · ·(A − λnI) Note that each matrix A − λiI is upper triangular. Now observe:
- A − λ 1 I has zero first column because column 1 of A is ( λ 1 , 0, 0,... , 0)T^.
- Then (A − λ 1 I)(A − λ 2 I) has the first two columns zero because the second column of (A − λ 2 I) is (b, 0, 0,... , 0)T^ for some constant b.
- Next (A − λ 1 I)(A − λ 2 I)(A − λ 3 I) has the first three columns zero because column 3 of (A − λ 3 I) is (c, d, 0,... , 0)T^ for some constants c and d.
Continuing in this way we see that (A − λ 1 I)(A − λ 2 I)(A − λ 3 I) · · ·(A − λnI) has all n columns zero; that is, cA(A) = 0.
Exercises for 8.
Exercise 8.7.1 In each case, compute the norm of the complex vector.
a. (1, 1 − i, −2, i)
b. ( 1 − i, 1 + i, 1, − 1 )
c. ( 2 + i, 1 − i, 2, 0, −i)
d. (−2, −i, 1 + i, 1 − i, 2i)
Exercise 8.7.2 In each case, determine whether the two vectors are orthogonal.
a. (4, − 3 i, 2 + i), (i, 2, 2 − 4 i)
b. (i, −i, 2 + i), (i, i, 2 − i)
c. (1, 1, i, i), (1, i, −i, 1)
d. ( 4 + 4 i, 2 + i, 2i), (− 1 + i, 2, 3 − 2 i)
Exercise 8.7.3 A subset U of Cn^ is called a complex subspace of Cn^ if it contains 0 and if, given v and w in U , both v + w and zv lie in U (z any complex number). In each case, determine whether U is a complex subspace of C^3.
a. U = {(w, w, 0) | w in C}
b. U = {(w, 2w, a) | w in C, a in R}
c. U = R^3
d. U = {(v + w, v − 2 w, v) | v, w in C}
Exercise 8.7.4 In each case, find a basis over C, and determine the dimension of the complex subspace U of C^3 (see the previous exercise).
a. U = {(w, v + w, v − iw) | v, w in C}
b. U = {(iv + w, 0, 2v − w) | v, w in C}
c. U = {(u, v, w) | iu − 3 v + ( 1 − i)w = 0; u, v, w in C}
d. U = {(u, v, w) | 2 u + ( 1 + i)v − iw = 0; u, v, w in C}
Exercise 8.7.5 In each case, determine whether the given matrix is hermitian, unitary, or normal. [ 1 −i i i
] a.
[ 2 3 − 3 2
] b. [ 1 i −i 2
] c.
[ 1 −i i − 1
] d.
√^1 2
[ 1 − 1 1 1
] e.
[ 1 1 + i 1 + i i
] f.
8.7. Complex Matrices 471
[ 1 + i 1 −i − 1 + i
] g. √ 21 |z|
[ z z z −z
] h. , z 6 = 0
Exercise 8.7.6 Show that a matrix N is normal if and only if NNT^ = NT^ N.
Exercise 8.7.7 Let A =
[ z v v w
] where v, w, and z are
complex numbers. Characterize in terms of v, w, and z when A is
a. hermitian b. unitary c. normal.
Exercise 8.7.8 In each case, find a unitary matrix U such that U H^ AU is diagonal.
a. A =
[ 1 i −i 1
]
b. A =
[ 4 3 − i 3 + i 1
]
c. A =
[ a b −b a
] ; a, b, real
d. A =
[ 2 1 + i 1 − i 3
]
e. A =
1 0 1 + i 0 2 0 1 − i 0 0
f. A =
1 0 0 0 1 1 + i 0 1 − i 2
Exercise 8.7.9 Show that 〈Ax, y〉 = 〈x, AH^ y〉 holds for all n × n matrices A and for all n-tuples x and y in Cn.
Exercise 8.7.
a. Prove (1) and (2) of Theorem 8.7.1.
b. Prove Theorem 8.7.2.
c. Prove Theorem 8.7.3.
Exercise 8.7.
a. Show that A is hermitian if and only if A = AT^.
b. Show that the diagonal entries of any hermitian matrix are real.
Exercise 8.7.
a. Show that every complex matrix Z can be written uniquely in the form Z = A + iB, where A and B are real matrices.
b. If Z = A + iB as in (a), show that Z is hermi- tian if and only if A is symmetric, and B is skew- symmetric (that is, BT^ = −B).
Exercise 8.7.13 If Z is any complex n × n matrix, show that ZZH^ and Z + ZH^ are hermitian.
Exercise 8.7.14 A complex matrix B is called skew- hermitian if BH^ = −B.
a. Show that Z −ZH^ is skew-hermitian for any square complex matrix Z.
b. If B is skew-hermitian, show that B^2 and iB are hermitian.
c. If B is skew-hermitian, show that the eigenvalues of B are pure imaginary (i λ for real λ ).
d. Show that every n × n complex matrix Z can be written uniquely as Z = A + B, where A is hermi- tian and B is skew-hermitian.
Exercise 8.7.15 Let U be a unitary matrix. Show that:
a. ‖U x‖ = ‖x‖ for all columns x in Cn.
b. | λ | = 1 for every eigenvalue λ of U.
