10.1.1 Properties of eigenvalues

In this section we will derive some useful properties of eigenvalues that follow directly from (10.3).

Firstly, if A is singular, that is, , then it follows from (10.3) that it has an eigenvalue λ = 0; conversely, if A has an eigenvalue λ = 0, then it is singular. Secondly, it follows from (10.4a) and (10.4c) that

Setting λ = 0 then gives

(10.5)

that is, the determinant of any matrix is equal to the product of its eigenvalues. Similarly, as we shall show, the sum of the eigenvalues is given by

(10.6)

where the trace is

(10.7)

Together with (10.6), Equation (10.7) is very useful in checking that the eigenvalues of a given matrix have been computed correctly. It is proved by computing the coefficient of λ^{n − 1} in (10.4b) using (10.4a) and (10.4c) in turn, and comparing the results. In (10.4a), the co-factors of a₁₂, a₁₃, …, a_1n are polynomials of order λ^{n − 2}. Hence terms of order λ^{n − 1} can only occur in the product of the diagonal elements in (10.4a), giving

On the other hand, expanding (10.4c) gives

and comparing the two expressions yields the desired result.

Finally, suppose that an n × n matrix A has k ≤ n distinct eigenvalues λ₁, λ₂, …, λ_k, that is, λ_i ≠ λ_j for i ≠ j and i, j ≤ k. Then the following related matrices also have a total of k distinct eigenvalues, as specified below.

1. The transpose matrix A^T has the same eigenvalues λ_i.
2. The matrix αA has eigenvalues αλ_i, where α is a scalar constant.
3. The Hermitian conjugate matrix A† has eigenvalues λ*_i.
4. The inverse matrix A^{− 1}, if it exists, has eigenvalues λ^{− 1}_i.

Here we will prove (iii) and leave the others as exercises for the reader. Since λ_i is an eigenvalue of A,

which by (9.59b) implies

From (9.33) and (9.55), we have

so that

and hence λ*_i is an eigenvalue of A† for all i = 1, 2, …, k. That they are the only distinct eigenvalues of A†, even if k < n, follows by using the argument in reverse. Suppose A† had an extra eigenvalue λ_i ≠ λ*_i, i = 1, 2, …, k. Then since (A†)† = A, this would imply that A had a distinct eigenvalue λ ≠ λ_i, i = 1, 2, …, k, in contradiction to the requirement that k is the total number of distinct eigenvalues of A.

10.1.2 Properties of eigenvectors

If x⁽ⁱ⁾ (i = 1, 2, …, k) is a set of eigenvectors corresponding to k different eigenvalues λ_i (i = 1, 2, …, k), then x⁽ⁱ⁾ are linearly independent. That is, there is no linear relationship of the type

(10.9)

where the c_i are constants, except the trivial case c_i = 0 where i = 1, 2, …, k. The proof is as follows.

Since Ax⁽ⁱ⁾ = λ_ix⁽ⁱ⁾ (i = 1, 2, …, k),

(10.10)

Suppose now that a condition of the form (10.9) does exist and we operate on it by (A − λ_jI), with the result

(10.11)

For j = 2, using (10.10) and (10.11) gives

(10.12)

where the term in x⁽²⁾ is absent. If this operation is now repeated on (10.12) using j = 3, an additional bracket (λ₁ − λ₃) multiplying each term will be generated and the term in x⁽³⁾ will be eliminated. Repeating the operation for the remaining values of j successively, eventually yields the result

and since all the λ_i are assumed to be different, this implies that c₁ = 0. The same method can be used to show that c₂ = 0, and so on. Hence if all the values of λ_i are different, only the trivial solution c_i = 0 (i = 1, 2, 3, …, k) exists, and so the eigenvectors are linearly independent.

We next consider the implications of this for an n × n matrix A. If all the eigenvalues λ_i (i = 1, 2, …, n) are distinct, then k = n above and there are n linearly independent eigenvectors x⁽¹⁾, x⁽²⁾, …, x⁽ⁿ⁾. Since an n-dimensional space cannot contain more than n linearly independent vectors, the eigenvectors form a complete set of linearly independent vectors, as defined in Section 9.2.1. Hence an arbitrary vector x can always be written as a sum of eigenvectors of the form

(10.13)

where the numerical constants α_i depend on x.

It remains to consider the case where k < n, that is, when there are less than n distinct eigenvalues. To illustrate this, suppose the characteristic polynomial is of the form

so that there are k = n − 1 distinct eigenvalues. Nonetheless, one can usually find two linearly independent eigenvectors x^{(n − 1)}, x⁽ⁿ⁾ that both have eigenvalue λ_{n − 1}. Hence there are still n linearly independent eigenvectors, and an arbitrary vector x can still be expanded in the form (10.13). However, sometimes, as we shall illustrate by an example below, there is only a single eigenvector x^{(n − 1)} corresponding to λ_{n − 1}. Hence there are only n − 1 linearly independent eigenvectors. Matrices like these, which have fewer independent eigenvectors than dimension of the matrix, are called defective matrices. For such matrices, an arbitrary vector in the n dimensional space cannot be expanded in terms of its eigenvectors.

10.1.3 Hermitian matrices

In most physical applications, and especially in quantum mechanics, the eigenvalues and eigenvectors of interest are those of Hermitian matrices. This is because the eigenvalues are real and so can correspond to measurable quantities. In addition, the eigenvectors corresponding to different eigenvalues are not only linearly independent, but also orthogonal. In particular, these results apply to real, symmetric matrices, which are automatically Hermitian.

To prove these properties, consider a Hermitian matrix A and an eigenvector a, corresponding to an eigenvalue λ_a, so that

(10.14a)

Taking the Hermitian conjugate, we obtain

(10.14b)

where we have used A = A† and the relation

which follows from (9.33) and (9.53). Then multiplying (10.14a) on the left by a† and (10.14b) on the right by a, we obtain

and

Since (a, a) ≠ 0, these equations can only be satisfied if λ_a = λ*_a, that is, the eigenvalue is real, as required.

Next we consider a second eigenvector b satisfying

(10.14c)

On multiplying (10.14c) on the left by a† and (10.14b) on the right by b, we obtain

and

where in the second equation we have used the result λ* = λ proved above. Since λ_a ≠ λ_b, these two equations are only compatible if

(10.15)

that is, the eigenvectors are orthogonal.

An n × n Hermitian matrix A always has n linearly independent eigenvectors⁴ x⁽ⁱ⁾. Hence an arbitrary n-dimensional vector can always be expanded in the form (10.13), that is,

(10.16a)

where

(10.16b)

and we have chosen unit eigenvectors . If the eigenvalues λ_i are all different, then the eigenvectors are orthonormal, that is,

(10.17a)

where δ_ij is the kronecker delta symbol defined in (9.24b). Multiplying (10.16a) by and using (10.17a) then gives

(10.17b)

for the coefficients α_j.

Equations (10.16) and (10.17) are very convenient in applications, but are only automatically valid if the eigenvalues λ_i are all different. If this is not so, the eigenvectors (10.16b) are not uniquely defined. However, one may always choose a complete set of linearly independent eigenvectors (10.16a) and (10.16b) that do satisfy (10.17a) and (10.17b). To see this, let us suppose there are k linearly independent eigenvectors u⁽¹⁾, u⁽²⁾, …, u^(k) corresponding to a given eigenvalue , that is,

Then the eigenvalue is said to be k-fold degenerate and any linear combination of the form

(10.18)

where the α_i are arbitrary constants, is also an eigenvector. In particular, it is possible to choose a sequence of eigenvectors

(10.19a)

in which each x⁽ⁱ⁾, i ≤ k, is chosen to be orthogonal to all x^(j) with j < i. These can then be normalised:

(10.19b)

This procedure is called Gram-Schmidt orthogonalisation, and the resulting eigenvectors x⁽ⁱ⁾ satisfy (10.17a), as required. They are, however, not unique and other choices of linearly independent eigenvectors satisfying (10.17a) are also possible.

*10.2.1 Normal modes of oscillation

In physical applications, diagonalisation of a matrix often enables one to choose a set of variables that decouple from each other. A typical application in mechanics is that of coupled oscillations. An example is given in Figure 10.1. This shows two equal masses m that are joined by a spring and suspended from fixed points by strings of equal length l. We will analyse the motion of the system when the weights are displaced small distances from their equilibrium positions, as shown.

If the instantaneous displacements are x₁ and x₂, then the force due to the spring pulling the two masses together is mk(x₂ − x₁), where mk is the spring constant. The tension T_i in the string produces a horizontal restoring force of magnitude mgx_i/l, for small displacements, and so the equations of motion of the system are

(10.25a)

and

(10.25b)

These coupled equations may be written in the matrix form

(10.26a)

where

(10.26b)

We now look for a transformation P such that

and

Since P is independent of t, the equations of motion become

so that in terms of x′₁ and x′₂, the equations of motion decouple

(10.27)

The eigenvalues are obtained using the characteristic equation

that is

The solution of the equations of motion (10.27) are then

(10.28a)

and

(10.28b)

where , , and where a₁, b₁, a₂, b₂ are arbitrary constants. If the latter are chosen such that x′₁ = 0 (or x′₂ = 0), the system vibrates with a single frequency ω₁ (or ω₂) and the motion is called a normal mode of the system. In general the actual motion will be a linear combination of its normal modes.

To express the motion (10.28) in terms of the original variables x₁, x₂, we need to find the matrix P. To do this, we first have to find the eigenvectors u⁽¹⁾ and u⁽²⁾. Using the techniques discussed previously, we find the two eigenvectors

Thus, from x = P x′,

(10.29)

which, together with (10.28), completes the matrix analysis of solution. Specific motions depend on the values of the constants a₁, b₁, a₂, b₂, as shown in Example 10.6 below.

Finally, we note that coupled oscillations occur in a wide variety of contexts in physical science, which include compound pendulums, electrical circuits and infra-red spectroscopy. Provided the oscillations are small,⁷ as in the example above, they are always described by equations of the form (10.26a), where A can in general be a real n × n matrix with n ≥ 2. As in the example, these are solved by diagonalising the matrix to obtain a set of n decoupled equations analogous to (10.27), with solutions of the form (10.28) for each of the new variables. Further examples, from classical mechanics, are explored in the problems at the end of this chapter.

*10.2.2 Quadratic forms

Another example of matrix diagonalisation occurs in the theory of quadratic forms. These are expressions of the type

(10.30)

where the quantities x_i and the coefficients a_ij are real. The latter form an n × n square matrix A, so (10.30) may be written

(10.31)

where x^T = (x₁, x₂, ⋅⋅⋅, x_n). Furthermore, it can be seen from (10.30) that Q is the sum of terms of the form (a_ij + a_ji)x_ix_j, which may be written (c_ijx_ix_j + c_jix_jx_i), where

Hence the quadratic form (10.31) can always be written in the form

where C is a real symmetric matrix. Therefore, in considering the quadratic forms (10.30), we may, without loss of generality, consider only cases where A is a real symmetric matrix. If Q > 0, it is said to be positive definite.

One application of quadratic forms is in analytic geometry. For example, suppose a surface in three-dimensional space is described by the equation

(10.32)

where x, y, z are Cartesian co-ordinates and k is a constant. Because of the cross terms in xy, etc., it is not obvious what is the geometrical nature of the surface. Its visualisation would be simpler if the surface could be expressed in co-ordinates such that the cross terms were absent. This may be done by using the technique of diagonalisation. We start by writing (10.32) in the matrix form

(10.33)

where x = (x, y, z)^T and A is a real symmetric matrix. Since A is Hermitian it can be diagonalised by a unitary matrix P, where P^{− 1} = P†; and since it is also real, it can be chosen to be a real orthogonal matrix, with P^{− 1} = P^T, so that

where λ_i (i = 1, 2, 3) are the eigenvalues of A. Given P, we can define new co-ordinates x′ = (x′, y′, z′) in terms of which (10.32) becomes simpler. The equation for the surface in these new co-ordinates may be found by writing

so that (10.33) becomes

(10.34)

where x′ = P^Tx = (x′, y′, z′). Writing this in terms of the new Cartesian co-ordinates gives

(10.35)

which is the equation of the quadratic surface where the eigenvectors of A define the direction the new co-ordinate axes x′, y′, z′, called the principal axes. They are related to the original axes x, y, z by rotations about, and possibly a reflection in, the origin.

The geometrical interpretation depends on the signs of the denominators in (10.35). If all three are positive, then (10.35) describes an ellipsoid, as shown in Figure 10.3. In this case the principal axis x′, for example, cuts the quadratic surface where y′ = z′ = 0, which from (10.35) is where x′ = ±(k/λ₁)^1/2. Thus the distance along the x′ axis from the origin to the point of intersection is a = (k/λ₁)^1/2. This is called the length of the semi-axis. The lengths of the other semi-axes are similarly given by b = (k/λ₂)^1/2 and c = (k/λ₃)^1/2, as shown in Figure 10.3.⁸

If all three denominators are different, then the ellipsoid is said to be triaxial. More familiar shapes are obtained when two of the denominators are equal. For example, if a = b > c, the ellipsoid reduces to an oblate spheroid, as shown in Figure 10.4b; while if a = b < c, it reduces to a prolate spheroid, as shown in Figure 10.4a. A familiar example of the former is the shape of earth, which is to a good approximation an oblate spheroid; while a rugby (or American) football is roughly a prolate spheroid. If a = b = c, the spheroid reduces to a sphere.

Finally, if one of the denominators in (10.35) is negative, the shape is a hyperboloid of one sheet, while if two are negative, it corresponds to a hyperboloid of two sheets, as shown in Figure 10.5a and Figure 10.5b, respectively. Examples of the former are the large cooling towers seen at power stations.