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CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems

Prerequisite for this chapter is some familiarity with the notion of a matrix and with the
two algebraic operations for matrices. Otherwise the chapter is independent of Chap. 7,
so that it can be used for teaching eigenvalue problems and their applications, without
first going through the material in Chap. 7.

SECTION 8.1. The Matrix Eigenvalue Problem. Determining Eigenvalues

and Eigenvectors, page 323
Purpose. To familiarize the student with the determination of eigenvalues and eigenvectors
of real matrices and to give a first impression of what one can expect (multiple eigenvalues,
complex eigenvalues, etc.).
Main Content, Important Concepts
Eigenvalue, eigenvector
Determination of eigenvalues from the characteristic equation
Determination of eigenvectors
Algebraic and geometric multiplicity, defect
Comments on Content
To maintain undivided attention on the basic concepts and techniques, all the examples in
this section are formal, and typical applications are put into a separate section (Sec. 8.2).
The distinction between the algebraic and geometric multiplicity is mentioned in this
early section, and the idea of a basis of eigenvectors (“eigenbasis”) could perhaps be
mentioned briefly in class, whereas a thorough discussion of this in a later section (Sec. 8.4)
will profit from the increased experience with eigenvalue problems, which the student
will have gained at that later time.
The possibility of normalizing any eigenvector is mentioned in connection with
Theorem 2, but this will be of greater interest to us only in connection with orthonormal
or unitary systems (Secs. 8.4 and 8.5).
In our present work we find eigenvalues first and are then left with the much simpler task
of determining corresponding eigenvectors. Numeric work (Secs. 20.62–20.9) may proceed
in the opposite order, but to mention this here would perhaps just confuse the student.
Further Comments on Examples and Theorems
The simple examples should give the student a first impression of what to expect.
In particular, Example 4 shows that eigenvalue problems lead to work in complex, even
if the matrices are real. This is an important point to emphasize.
Theorem 1 shows that for matrices, in contrast to differential equations, eigenvalue
problems involve no existence questions since the existence of an eigenvalue is always
guaranteed.
Theorems 2 and 3 concern the notion of eigenspace and the invariance of eigenvalues
under transposition.
Comments on Problems
Problems 1–16 involve straightforward calculations to gain skill and an understanding of
the concepts involved. Sidetracking attention by solving cubic or higher-order equations
is avoided.

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Problems 17–20 illustrate simple applications to analytic geometry. Actual applications

of eigenvalue problems follow in the next section, as has been mentioned before.
Problems 21–25 illustrate some important simple facts.

SOLUTIONS TO PROBLEM SET 8.1, page 329

1. Eigenvalues are: 3>2 and 3 and the corresponding eigenvectors are [1, 0]T, and [0, 1]T
respectively.
2. This zero matrix, like any square zero matrix, has the eigenvalue 0. The algebraic
multiplicity and geometric multiplicity are both equal to 2, and we can choose
the basis [1 0]T, [0 1]T.
3. Eigenvalues are 0 and ⫺3, and the corresponding eigenvectors are [2>3, 1]T, and
[1>3, 1]T respectively.
4. Eigenvalues are 5 and 0 and eigenvectors [1 2]T and [⫺2 1]T, respectively. The
matrix is symmetric, and for such a matrix it is typical that the eigenvalues are real
and the eigenvectors orthogonal. Also, make the students aware of the fact that 0 can
very well be an eigenvalue.
5. Eigenvalues are 4i and ⫺4i, and the corresponding eigenvectors are [⫺i, 1]T and
[i, 1]T respectively.
6. Eigenvalues are 1 and 3 and eigenvectors [1 0]T and [1 1]T, respectively. Note
that for such a triangular matrix, the main diagonal entries are still the eigenvalues
(as for a diagonal matrix; cf. Prob. 1), but the eigenvectors are no longer orthogonal.
7. The matrix has a repeated eigenvalue of 0 with eigenvectors [1, 0]T, and [0, 0]T.
8. Eigenvalues: a ⫾ 2⫺k; Eigenvectors: [1> 2⫺k, 1]T and [⫺1> 2⫺k, 1]T,
respectively.
9. Eigenvalues are 0.20 ⫾ 0.40, and the eigenvectors are [i, 1]T and [⫺i, 1]T
respectively.
10. The characteristic equation is
(cos u ⫺ l)2 ⫹ sin2 u ⫽ 0.
Solutions (eigenvalues) are l ⫽ cos u ⫾ i sin u. Eigenvectors are obtained from
(l ⫺ cos u)x 1 ⫹ (sin u)x 2 ⫽ (sin u)(⫾ix 1 ⫹ x 2) ⫽ 0,
say, x 1 ⫽ 1, x 2 ⫽ ⫿i.
Note that this matrix represents a rotation through an angle u, and this linear
transformation preserves no real direction in the x 1x 2-plane, as would be the case if
the eigenvectors were positive real. This explains why these vectors must be complex.
11. Eigenvalues: 4, 1, 7, Eigenvectors are: [⫺1>2, 1, 1]T, [1, ⫺1>2, 1]T, [⫺2, ⫺2, 1]T.
12. 3, [1 0 0]T; 4, [5 1 0]T; 1, [7, ⫺4 2]T
13. Repeated eigenvalue 2. Eigenvectors: [2, ⫺2, 1]T, [0, 0, 0]T, [0, 0, 0]T.
14. Develop the characteristic determinant by the second row, obtaining
(12 ⫺ l)[(2 ⫺ l)(4 ⫺ l) ⫹ 1] ⫽ (12 ⫺ l)(l ⫺ 3)2.
Eigenvectors for the eigenvalues 12 and 3 are [0 1 0]T and [⫺1 0 1]T,
respectively, and we get no basis for R3.
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16. The indicated division of the characteristic polynomial gives

(l4 ⫺ 22l2 ⫹ 24l ⫹ 45)>(l ⫺ 3)2 ⫽ l2 ⫹ 6l ⫹ 5.
The eigenvalues and eigenvectors are
l1 ⫽ 3, [1 1 1 1]T with a defect of 1
l2 ⫽ ⫺1 [3 ⫺1 1 1]T
l3 ⫽ ⫺5, [⫺11 1 5 1]T.

c d; c d; c d . (a) Any point of the x 1-axis is mapped onto

1 0 1 0
18. (a) 1, (b) ⫺1,
0 ⫺1 0 1
itself. (b) Any point (0, x 2) on the x 2-axis is mapped onto (0, ⫺x 2), so that [0 x 2]T
is an eigenvector corresponding to l ⫽ ⫺1.
1 1
2 2 0 0 1
20. D 12 1
2 0T . The eigenvalue 1 with eigenvectors D0T and D1T (which span the plane

0 0 1 1 0
x 2 ⫽ x 1) indicates that every point in the plane x 2 ⫽ x 1 is mapped onto itself. The
other eigenvalue 0 with eigenvector [1 ⫺1 0]T indicates that any point on the line
x 2 ⫽ ⫺x 1, x 3 ⫽ 0 (which is perpendicular to the plane x 2 ⫽ x 1) is mapped onto the
origin. The student should perhaps make a sketch to see what is going on geometrically.
24. By Theorem 1 in Sec. 7.8 the inverse exists if and only if det A ⫽ 0. On the other
hand, from the product representation
D(l) ⫽ det (A ⫺ lI) ⫽ (⫺1)n(l ⫺ l1)(l ⫺ l2) Á (l ⫺ ln)
of the characteristic polynomial we obtain
det A ⫽ (⫺1)n(⫺l1)(⫺l2) Á (⫺ln) ⫽ l1l2 Á ln.
Hence Aⴚ1 exists if and only if 0 is not an eigenvalue of A.
Furthermore, let l ⫽ 0 be an eigenvalue of A. Then
Ax ⫽ lx.
Multiply this by Aⴚ1 from the left:
Aⴚ1Ax ⫽ lAⴚ1x.
Now divide by l:
1
x ⫽ Aⴚ1x.
l

SECTION 8.2. Some Applications of Eigenvalue Problems, page 329

Purpose. Matrix eigenvalue problems are of greatest importance in physics, engineering,
geometry, etc., and the applications in this section and in the problem set are supposed to
give the student at least some impression of this fact.
Main Content
Applications of eigenvalue problems in
Elasticity theory (Example 1)
Probability theory (Example 2)
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Biology (Example 3)
Mechanical vibrations (Example 4)
Short Courses. Of course, this section can be omitted, for reasons of time, or one or two
of the examples can be considered quite briefly.
Comments on Content
The examples in this section have been selected from the viewpoint of modest
prerequisites, so that not too much time will be needed to set the scene.
Example 4 illustrates why real matrices can have complex eigenvalues (as
mentioned before, in Sec. 8.1), and why these eigenvalues are physically meaningful.
(For students familiar with systems of ODEs, one can easily pick further examples
from Chap. 4.)
Comments on Problems
Problems 1–12 are similar to the applications shown in the examples of the text.
Problems 13–15 show an interesting application of eigenvalue problems to production,
typical of various other applications of eigenvalue theory in economics included in various
textbooks in economic theory.

SOLUTIONS TO PROBLEM SET 8.2, page 333

1. Eigenvalue and eigenvectors are ⫺1, [⫺1, 1]T and 2, [1, 1]T. The eigenvectors are
orthogonal.
2. Eigenvalues and eigenvectors are 1.6, [1 ⫺1]T and 2.4, [1 1]T. These vectors are
orthogonal, as is typical of a symmetric matrix. Directions are ⫺45° and 45°,
respectively.
3. Eigenvalues 3, ⫺3 and eigenvectors [22, 1]T and [⫺1> 22, 1]T respectively.
4. Extension factors 9 ⫹ 215 ⫽ 13.47 and 9 ⫺ 215 ⫽ 4.53 in the directions given by
[1 2 ⫹ 15]T and [1 2 ⫺ 15]T (76.7° and ⫺13.3°, respectively).
1
6. 2, [1 1]T; 2, [1 ⫺1]T; directions 45° and ⫺45°, respectively.
7. Eigenvector [2.5, 1]T with eigenvalue 1.
8. [1 1 1]T, as could also be seen without calculation because A has row sums equal
to 1, which would not be the case in general.
9. Eigenvector [⫺0.2, ⫺0.4, 1] with eigenvalue 1.
10. Growth rate 3. The characteristic polynomial is 14(x ⫺ 3)(2x ⫹ 5)(2x ⫹ 1) which gives
the remaining two eigenvalues as 2.5 and 0.5. The sum of all the eigenvalues is the
trace which is zero. Note that the growth rate is not that sensitive to the elements of
the matrix. Working with two decimal digits still retains the intrinsic characteristic
of the problem.
11. Growth rate is 4. Characteristic polynomial is (x ⫺ 4)(x ⫹ 1)(x ⫹ 3).
12. Growth rate 1.3748. The other eigenvalues are complex or negative and are not needed.
The sum of all eigenvalues equals the trace, that is, 0, except for a roundoff error.
This 4 ⫻ 4 Leslie matrix corresponds to a classification of the population into four
classes.
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14. A has the same eigenvalues as AT, and AT has row sums 1, so that it has the eigenvalue
1 with eigenvector x ⫽ [1 Á 1]T.
Leontief is a leader in the development and application of quantitative methods in
empirical economical research, using genuine data from the economy of the United
States to provide, in addition to the “closed model” of Prob. 13 (where the producers
consume the whole production), “open models” of various situations of production
and consumption, including import, export, taxes, capital gains and losses, etc. See
W. W. Leontief, The Structure of the American Economy 1919–1939 (Oxford: Oxford
University Press, 1951). H. B. Cheney and P. G. Clark, Interindustry Economics (New
York: Wiley, 1959).
16. This follows by comparing the coefficient of lnⴚ1 in the development of the
characteristic determinant D (l) with that obtained from the product representation.
18. The first statement follows from
Ax ⫽ lx, (kA)x ⫽ k(Ax) ⫽ k(lx) ⫽ (kl)x,
the second by induction and multiplication of Akxj ⫽ lkj xj by A from the left.
20. det (L ⫺ lI) ⫽ ⫺l3 ⫹ l 12l 21l ⫹ l 13l 21l 32 ⫽ 0. Hence l ⫽ 0. If all three eigenvalues
are real, at least one is positive since trace L ⫽ 0. The only other possibility is
l1 ⫽ a ⫹ ib, l2 ⫽ a ⫺ ib, l3 real (except for the numbering of the eigenvalues). Then
l3 ⬎ 0 because
l1l2l3 ⫽ (a 2 ⫹ b 2)l3 ⫽ det L ⫽ l 13l 21l 32 ⬎ 0.

SECTION 8.3. Symmetric, Skew-Symmetric, and Orthogonal Matrices,

page 334
Purpose. To introduce the student to the three most important classes of real square
matrices and their general properties and eigenvalue theory.
Main Content, Important Concepts
The eigenvalues of a symmetric matrix are real.
The eigenvalues of a skew-symmetric matrix are pure imaginary or zero.
The eigenvalues of an orthogonal matrix have absolute value 1.
Further properties of orthogonal matrices.
Comments on Content
The student should memorize the preceding three statements on the locations of
eigenvalues as well as the basic properties of orthogonal matrices (orthonormality of row
vectors and of column vectors, invariance of inner product, determinant equal to 1 or ⫺1).
Furthermore, it may be good to emphasize that, since the eigenvalues of an orthogonal
matrix may be complex, so may be the eigenvectors. Similarly for skew-symmetric
matrices. Both cases are simultaneously illustrated by

c d c d c d
0 1 1 1
A⫽ with eigenvectors and
⫺1 0 i ⫺i
corresponding to the eigenvalues i and ⫺i, respectively.
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Further Comments on the Three Classes of Matrices in This Section

Reality of eigenvalues is a main reason for the importance of symmetric matrices—many
quantities in physics, such as mass, energy, etc., are real.
Formula (4) brings in skew-symmetric matrices in a rather natural fashion.
Theorem 3 explains the importance of orthogonal matrices.
Typical examples of the spectra of the matrices considered in this section are illustrated
by Probs. 1–10––most importantly, Probs. 4 and 8.
Problems 13–20 should help the student gain a deeper understanding of the concepts
and properties of the three classes of matrices considered in this section.

SOLUTIONS TO PROBLEM SET 8.3, page 338

1. Eigenvalues: 3>5 ⫾ 4>5i, Eigenvectors: [i, 1]T and [⫺i, 1]T respectively. Skew-
symmetric and orthogonal.
2. Eigenvalues a ⫾ ib. Symmetric if b ⫽ 0; then the eigenvalues are real. Skew-symmetric
if a ⫽ 0; then the eigenvalues are pure imaginary (or zero). Orthogonal if a 2 ⫹ b 2 ⫽ 1;
then the eigenvalues have an absolute value of 1.
3. Non-orthogonal; skew-symmetric; Eigenvalues 1 ⫾ 4i with eigenvectors [⫺i, 1]T
and [i, 1]T.
4. The characteristic equation is

cos u ⫺ l ⫺sin u
2 2 ⫽ l2 ⫺ (2 cos u) l ⫹ 1 ⫽ 0.
sin u cos u ⫺ l
Hence the eigenvalues are

l ⫽ cos u ⫾ i sin u.

If u ⫽ 0 (the identity transformation) we have l ⫽ 1 with multiplicity 2.

If u ⫽ 0, we obtain the eigenvectors from

x 2 ⫽ ⫿ix 1, say, [1 ⫿i],

which are complex; indeed, no (real) direction is preserved under a rotation.

5. Symmetric with eigenvalues 2, ⫺2, 3 and eigenvectors [0, 1> 23, 1]T, [0,⫺23, 1]T,
[1, 0, 0]T respectively; non-orthogonal.
6. a ⫹ 2k, [1 1 1]T; a ⫺ k, [1 0 ⫺1]T, [1 ⫺1 0]T; symmetric (for real a
and k)
7. Skew-symmetric; Eigenvalues: 0, ⫾32i with eigenvectors [⫺1, ⫺1>2, 1]T, [4>5
⫹ 3>5i, 2>5 ⫺ 6>5i, 1]T, [4>5 ⫺ 3>5i, 2>5 ⫹ 6>5i, 1]T respectively; non-orthogonal.
8. Orthogonal, a rotation about the x 1-axis through an angle u. Eigenvalues 1 and
cos u ⫾ i sin u. Compare with Prob. 4.
9. Skew-symmetric; Eigenvalues: ⫺1, and ⫾i with eigenvectors [0, 1, 0]T, [i, 0, 1]T,
[⫺i, 0, 1]T respectively; orthogonal.
10. Orthogonal; eigenvalues: ⫺1, 1, 1, all of absolute value 1. Eigenvectors [1, ⫺1, 1]T,
[⫺1, 0, 1]T, [1, 1, 0]T.
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12. CAS Experiment

(a) AT ⫽ Aⴚ1, BT ⫽ Bⴚ1, (AB)T ⫽ BTAT ⫽ Bⴚ1Aⴚ1 ⫽ (AB)ⴚ1. Also (Aⴚ1)T ⫽
(AT)ⴚ1 ⫽ (Aⴚ1)ⴚ1. In terms of rotations it means that the composite of rotations and
the inverse of a rotation are rotations.
(b) The inverse is

c d.
cos u sin u

⫺sin u cos u
(c) To a rotation of about 36.87°. No limit. For a student unfamiliar with complex
numbers this may require some thought.
(d) Limit 0, approach along some spiral.
(e) The matrix is obtained by using familiar values of cosine and sine,

c d.
13>2 ⫺12
A⫽
1
2 13>2
16. Let Ax ⫽ lx (x ⫽ 0), Ay ⫽ ␮y (y ⫽ 0). Then
lx T ⫽ (Ax)T ⫽ x TAT ⫽ x TA.
Thus
lx Ty ⫽ x TAy ⫽ x T␮y ⫽ ␮x Ty.
Hence if l ⫽ ␮, then x Ty ⫽ 0, which proves orthogonality.
18. det A ⫽ det (AT) ⫽ det (⫺A) ⫽ (⫺1)n det A ⫽ ⫺det A ⫽ 0 if n is odd. Hence the
answer is no. For even n ⫽ 2, 4, Á we have

0 1 0 0

c d,
0 1 ⫺1 0 0 0
E U, etc,
⫺1 0 0 0 0 1

0 0 ⫺1 0
20. Yes, for instance,
1
2 13>2 0
D 13>2 ⫺12 0T .

0 0 1

SECTION 8.4. Eigenbases. Diagonalization. Quadratic Forms, page 339

Purpose. This section exhibits the role of bases of eigenvectors (“eigenbases”) in
connection with linear transformations and contains theorems of great practical importance
in connection with eigenvalue problems.
Main Content, Important Concepts
Bases of eigenvectors (Theorems 1, 2)
Similar matrices have the same spectrum (Theorem 3)
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Diagonalization of matrices (Theorem 4)

Principal axes transformation of forms (Theorem 5)
Short Courses. Complete omission of this section or restriction to a short look at Theorems
1 and 5.
Comments on Content
Theorem 1 on similar matrices has various applications in the design of numeric methods
(Chap. 20), which often use subsequent similarity transformations to tridiagonalize or
(nearly) diagonalize matrices on the way to approximations of eigenvalues and
eigenvectors. The matrix X of eigenvectors [see (5)] also occurs quite frequently in that
context.
Theorem 2 is another result of fundamental importance in many applications, for
instance, in those methods for numerically determining eigenvalues and eigenvectors. Its
proof is substantially more difficult than the proofs given in this chapter.
The theorems in this section give sufficient conditions for the existence of eigenbases
(⫽ bases of eigenvectors), namely, the almost trivial Theorem 1 as well as the very
important Theorem 2, exhibiting another basic property of symmetric matrices.
This is followed in Theorems 3 and 4 by similarity of matrices and its application to
diagonalization.
The second part of the section concerns the principal axes transformation of quadratic
forms and its application to conic sections.
The extension of these ideas and results to complex matrices and forms follows in the
next section, the last one of this chapter.

SOLUTIONS TO PROBLEM SET 8.4, page 345

c d; c d; c d.
⫺11
9
1
18 0 ⫺1
2. Â ⫽ l ⫽ ⫺1, y ⫽ x ⫽ Py ⫽
⫺80
9
11
9 1 ⫺4
Similarly, for the second eigenvalue we obtain

c d; c d.
2>9 ⫺1>9
l ⫽ 1, y ⫽ x ⫽ Py ⫽
1 ⫺40>9

15 0 26 l ⫽ ⫺1, y ⫽ [26 1 ⫺16]T, x ⫽ [4 1 ⫺2]T

4. Â ⫽ D 6 3 10T ; l ⫽ 3, y ⫽ [0 1 0]T, x ⫽ [0 1 0]T

⫺8 0 ⫺14 l ⫽ 2, y ⫽ [⫺2 2 1]T, x ⫽ [⫺1 2 ⫺1]T

4 ⫺2 4
5. ˆ ⫽ D0
A ⫺2 12T;

0 ⫺2 12
l ⫽ 10, y ⫽ [1>3, 1, 1]T, x ⫽ Py ⫽ [1, 1>3, 1]T
l ⫽ 4, y ⫽ [1, 0, 0]T, x ⫽ Py ⫽ [0, 1, 0]T
l ⫽ 0, y ⫽ [2, 6, 1]T, x ⫽ Py ⫽ [6, 2, 1]T
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6. Project. (a) This follows immediately from the product representation of the
characteristic polynomial of A.
n n
(b) C ⫽ AB, c11 ⫽ a a1lbl1, c22 ⫽ a a2lbl2, etc. Now take the sum of these n sums.
l⫽1 l⫽1
Furthermore, trace BA is the sum of
n n
~
c 11 ⫽ a b1mam1, Á , ~
cnn ⫽ a bnmamn,
m⫽1 m⫽1
2
involving the same n terms as those in the double sum of trace AB.
(c) By multiplications from the right and from the left we readily obtain
~
A ⫽ P 2ÂP ⫺2.
(d) Interchange the corresponding eigenvectors (columns) in the matrix X in (5).
⫺1 1
9. Eigenvalues: ⫺2, 6; Matrix of corresponding eigenvectors: C S
1 1
10. The eigenvalues of A are ⫺1 and 1. A matrix of corresponding eigenvectors is

c d.
0 1>2
X⫽
1 1

c d.
⫺2 1
Xⴚ1 ⫽
0 1
Hence diagonalization gives

c d.
⫺1 0
D ⫽ Xⴚ1AX ⫽
0 1

12. A has the eigenvalues 10 and ⫺5. A matrix of eigenvectors is

c d.
7 11
X⫽
13 ⫺1
Its inverse is

c d.
1>150 11>150
Xⴚ1 ⫽
13>150 ⫺7>150
Hence diagonalization gives

c d.
10 0
D ⫽ Xⴚ1AX ⫽
0 ⫺5
⫺18
7 6>7 0
25
13. Eigenvalues: 2, 9, 6; Matrix of corresponding eigenvectors: D 7 ⫺6>7 1T

⫺1 0 0
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14. A has the eigenvalues ⫺2, 4, 1. A matrix of eigenvectors is

2 0 ⫺1

X ⫽ D1 1 1T .

2 1 0
Its inverse is

1 1 ⫺1

Xⴚ1 ⫽ D⫺2 ⫺2 3T .

1 2 ⫺2
Hence diagonalization gives

⫺2 0 0

D ⫽ Xⴚ1AX ⫽ D 0 4 0T .

0 0 1
0 ⫺3 ⫺1>2

15. Eigenvalues: 5, ⫺1, 3; Matrix of corresponding eigenvectors: D1 1 0T

1 1 1
16. A has eigenvalues ⫺2, 2, 0. A matrix of corresponding eigenvectors is

1 0 ⫺1

X ⫽ D1 0 1T .

0 1 0
Its inverse is

1>2 1>2 0

Xⴚ1 ⫽ D 0 0 1T .

⫺1>2 1>2 0
Diagonalization thus gives

⫺2 0 0

D ⫽ Xⴚ1Ax ⫽ D 0 2 0T .

0 0 0
18. The symmetric coefficient matrix is

c d.
4 3
C⫽
3 ⫺4
It has eigenvalues 5 and ⫺5. Hence the transformed quadratic form is
5y 21 ⫺ 5y 22 ⫽ 10, or y 21 ⫺ y 22 ⫽ 2.
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This is a hyperbola. The matrix whose columns are normalized eigenvectors of C

gives the relation between y and x in the form

c d y.
3>10110 ⫺1>10110
x⫽
1>10110 3>10110
20. The symmetric coefficient matrix is

c d.
9 3
C⫽
3 1
Its eigenvalues are 0 and 10. Hence the transformed form is
10y 22 ⫽ 10.
This represents a pair of parallel straight lines
10y 22 ⫽ 10, thus y2 ⫽ ⫾1.
The matrix X whose columns are normalized eigenvectors of C gives the relation
between y and x in the form

c d y.
1 1 3
x⫽
110 ⫺3 1

22. The symmetric coefficient matrix is

c d.
4 16
C⫽
6 13
Its eigenvalues are 1 and 16. Hence the transformed form is
y 21 ⫹ 16y 22 ⫽ 16.
This represents an ellipse. The matrix whose columns are normalized eigenvectors of
C gives the relation between y and x in the form

c d y.
1 ⫺2 1
x⫽
15 1 2

24. Transform Q (x) by (9) to the canonical form (10). Since the inverse transform
y ⫽ X⫺1x of (9) exists, there is a one-to-one correspondence between all x ⫽ 0 and
y ⫽ 0. Hence the values of Q (x) for x ⫽ 0 coincide with the values of (10) on the
right. But the latter are obviously controlled by the signs of the eigenvalues in the
three ways stated in the theorem. This completes the proof.

SECTION 8.5. Complex Matrices and Forms. Optional, page 346

Purpose. This section is devoted to the three most important classes of complex matrices
and corresponding forms and eigenvalue theory.
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Main Content, Important Concepts

Hermitian and skew-Hermitian matrices
Unitary matrices, unitary systems
Location of eigenvalues (Fig. 163)
Quadratic forms, their symmetric coefficient matrix
Hermitian and skew-Hermitian forms
Background Material. Section 8.3, which the present section generalizes. The
prerequisites on complex numbers are very modest, so that students will need hardly any
extra help in that respect.
Short Courses. This section can be omitted.
The importance of these matrices results from quantum mechanics as well as from
mathematics itself (e.g., from unitary transformations, product representations of
nonsingular matrices A ⫽ UH, U unitary, H Hermitian, etc.).
The determinant of a unitary matrix (see Theorem 4) may be complex. For example,
the matrix

c d
1⫹i 1 0
A⫽
12 0 1
is unitary and has
det A ⫽ i.

Comments on Problems
Complex matrices appear in quantum mechanics; see Prob. 7, etc.
Problems 13–20 give an impression of calculations for complex matrices.
Normal matrices, defined in Prob.18, play an important role in a more extended theory
of complex matrices.

SOLUTIONS TO PROBLEM SET 8.5, page 351

1. Hermitian; Eigenvalues: 3, 1; and the corresponding matrix of eigenvectors is

i ⫺i
C S
1 1
2. Skew-Hermitian. Eigenvalues and eigenvectors are
⫺i, [⫺1 ⫹ i 2]T
and
2i, [2 1 ⫹ i]T.

1 ⫺1
3. Non-Hermitian; Eigenvalues: 14 ⫾ i22, and the matrix of eigenvectors is C S
1 1
4. Skew-Hermitian, as well as unitary, eigenvalues i and ⫺i, eigenvectors [1 1]T and
[1 ⫺1]T, respectively.
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5. Non-Hermitian; Eigenvalues: i, ⫺i, ⫺i, and the corresponding matrix of eigenvectors is

0 1 0

D1 0 0T

0 0 1
6. Hermitian. Eigenvalues and eigenvectors are
⫺4, [i ⫺1 ⫺ i 1]T
0, [1 0 i]T
4, [i 1⫹i 1]T.
8. Eigenvectors are as follows. (Multiplication by a complex constant may change them
drastically!)
For A [1 ⫺ 3i 5]T, [1 ⫺ 3i ⫺2]T
For B [2 ⫹ i i]T, [2 ⫹ i ⫺5i]T
For C [1 1]T, [1 ⫺1]T.
9. Skew-Hermitian; x TAx ⫽ 8 ⫺ 12i.
10. The matrix is non-Hermitian.
x TAx ⫽ [3, ⫺2i][⫺4 ⫹ i, 3 ⫹ 6i]T ⫽ ⫺3i

11. Skew-Hermitian; x TAx ⫽ ⫺4i.

12. The matrix is Hermitian. We obtain the real value
[1 ⫺i i]A[1 i ⫺i]T ⫽ [1 ⫺i i][⫺4i 2i 4 ⫺ 2i]T ⫽ 4.

14. (BA)T ⫽ (BA)T ⫽ A T B T ⫽ A(⫺B) ⫽ ⫺AB. For the matrices in Example 2.

c d.
1 ⫹ 19i 5 ⫹ 3i
AB ⫽
⫺23 ⫹ 10i ⫺1

16. The inverse of a product UV of unitary matrices is

(UV)ⴚ1 ⫽ Vⴚ1U ⴚ1 ⫽ V T U T ⫽ (UV )T.
This proves that UV is unitary.
We show that the inverse Aⴚ1 ⫽ B of a unitary matrix A is unitary. We obtain
Bⴚ1 ⫽ (Aⴚ1)ⴚ1 ⫽ (A T)ⴚ1 ⫽ (Aⴚ1)T ⫽ B T,
as had to be shown.
18. A TA ⫽ A2 ⫽ AA T if A is Hermitian, A TA ⫽ ⫺A2 ⫽ A(⫺A) ⫽ AA T if A is skew-
Hermitian, A TA ⫽ A⫺1A ⫽ I ⫽ AA⫺1 ⫽ AA T if A is unitary.
20. For instance,

c d
0 0

i 0
is not normal. A normal matrix that is not Hermitian, skew-Hermitian, or unitary is
obtained if we take a unitary matrix and multiply it by 2 or some other real factor
different from ⫾1.
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⫺1 1
11. Eigenvalues: 1, 2, and the corresponding matrix of eigenvectors is C S
1 1
12. The eigenvalues are ⫺1 and 1. Corresponding eigenvectors are [2 3]T and [1 2]T,
respectively. Note that this basis is not orthogonal.
2>5 2>3
13. Eigenvalues: 11/2, 13/2, and the matrix of eigenvectors is C S
1 1
14. One of the eigenvalues is 9. Its algebraic and geometric multiplicities are 2.
Corresponding linearly independent eigenvectors are [1 0 ⫺2]T and [0 1 2]T.
The other eigenvalue is 4.5. A corresponding eigenvector is [2 ⫺2 1]T.
15. Eigenvalues: ⫾12i, 0, and the matrix of eigenvectors is

⫺1>4 ⫹ 3>4i ⫺1>4 ⫺ 3>4i 2

D 1>4 ⫹ 3>4i 1>4 ⫺ 3>4i ⫺2T

1 1 1

16. The eigenvalues of A are ⫺17 and 9. The similar matrix, having the same
eigenvalues, is

c dc d c d
1>2 1>2 9 17 9 0
 ⫽ P⫺1AP ⫽ ⫽
⫺1>2 1>2 9 ⫺17 0 ⫺17

c 27 d
35
2 ⫺35
2
17. Eigenvalues: ⫺5 and 7. Â ⫽
⫺312 2
18. A has the eigenvalues ⫺2, ⫺1, 2. The inverse of P is

1 ⫺8 31
P ⴚ1 ⫽ D0 1 ⫺3T .

0 0 1
The similar matrix Â, having the same eigenvalues, is
1 ⫺8 31 ⫺4 ⫺26 52 ⫺35 ⫺259 345

 ⫽ D0 1 ⫺3T D 0 2 6T ⫽ D 3 23 ⫺27T .

0 0 1 ⫺1 ⫺7 11 ⫺1 ⫺7 11

19. Eigenvalues: 2>5 and ⫺1>20; The corresponding matrix of eigenvectors is

4>5 5>4 ⫺20

9
25
9
C S and its inverse is C S
20
1 1 9 ⫺16
9
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20. Eigenvalues are 26 and 316. The corresponding matrix of eigenvalues is

c d.
⫺1>17 17
X⫽
1 1
Note that the vectors are orthogonal. Its inverse is

c d.
17 289
⫺290 290
Xⴚ1 ⫽
17 1
290 290

Diagonalization gives

c dc d c d.
17 289
⫺290 290 ⫺316
17 442 316 0
⫽
17 1
290 290 316 26 0 26
22. The symmetric coefficient matrix is

c d.
9 ⫺3
C⫽
⫺3 17
Its eigenvalues are 8 and 18; they are both positive real. The transformed form is
8y 21 ⫹ 18y 22 ⫽ 36.
This is the canonical form; there is no y1y2-term. It represents an ellipse. The matrix
X whose columns are normalized eigenvectors of C gives the relationship between y
and x in the form

c d y.
1 3 1
x⫽
110 1 ⫺3
24. The symmetric coefficient matrix is

c d.
6 8
C⫽
8 ⫺6
Its eigenvalues are 10 and ⫺10. The transformed form is
10y 21 ⫺ 10y 22 ⫽ 10(y1 ⫹ y2)(y1 ⫺ y2) ⫽ 0.
It represents two perpendicular straight lines through the origin. The matrix X whose
columns are normalized eigenvectors of C gives the relationship between x and y in
the form

c d y.
2>525 ⫺1>525
x⫽
1>525 2>525