9.1 Matrix Multiplication

Comparing (9.4) with (9.5), it is seen that matrix multiplication implies the following rule involving their component elements:

(9.6)

Note that summation over identical adjacent indices results in their mutual “annihilation.” Suppose the quantities in Eq. (9.4) are themselves determined by simultaneous relations

(9.7)

The combined results of Eqs. (9.4) and (9.7), equivalent to eliminating between the two sets of equations, can be written

(9.8)

We can write the same equations in matrix notation:

(9.9)

Evidently, can be represented as a matrix product:

(9.10)

An element of the product matrix is constructed by summation over two sets of matrix elements in the following pattern:

(9.11)

The diagram below shows schematically how the ijth element is constructed from the sum of products of elements from the ith row of the first matrix and the jth column of the second:

The three Pauli spin matrices

(9.12)

will provide computationally simple examples to illustrate many of the properties of matrices. They are themselves of major significance in applications to quantum mechanics and geometry.

The most dramatic contrast between multiplication of matrices and multiplication of numbers is that matrix multiplication can be noncommutative, meaning that it is not necessarily true that

(9.13)

As a simple illustration, consider products of Pauli spin matrices: We find

(9.14)

also

(9.15)

Matrix multiplication remains associative, however, so that

(9.16)

In matrix multiplication, the product of an matrix and an matrix is an matrix. Two matrices cannot be multiplied unless their adjacent dimensions— in the above example—match. As we have seen above, square matrix multiplying a column vector gives another column vector (). The product of a row vector and a column vector is an ordinary number (in a sense, a matrix). For example,

(9.17)

9.2 Further Properties of Matrices

Following are a few hints on how to manipulate indices in matrix elements. It is most important to recognize that any index that is summed over is a dummy index. The result is independent of what we call it. Thus

(9.18)

Secondly, it is advisable to use different indices when a product of summations occurs in an expression. For example,

This becomes mandatory if we reexpress it as a double summation

Multiplication of a matrix by a constant is equivalent to multiplying each by . Two matrices of the same dimension can be added element by element. By combination of these two operations, the matrix elements of are given by .

The null matrix has all its elements equal to zero:

(9.19)

As expected,

(9.20)

A diagonal matrix has only nonvanishing elements along the main diagonal, for example

(9.21)

Its elements can be written in terms of the Kronecker delta:

(9.22)

A diagonal matrix is sometimes represented in a form such as

(9.23)

A special case is the unit or identity matrix, diagonal with all elements equal to 1:

(9.24)

Clearly,

(9.25)

As expected, for an arbitrary matrix :

(9.26)

9.3 Determinants

Determinants, an important adjunct to matrices, can be introduced as a geometrical construct. Consider the parallelogram shown in Figure 9.1, with one vertex at the origin and the other three at , , and . Using Pythagoras’ theorem, the two sides and the diagonal have the lengths

(9.27)

The area of the parallelogram is given by

(9.28)

where is the angle between sides and . The sign is determined by the relative orientation of and . Also, by the law of cosines,

(9.29)

Eliminating between Eqs. (9.28) and (9.29), we find, after some lengthy algebra, that

(9.30)

(If you know about the cross product of vectors, this follows directly from .) This combination of variables has the form of a determinant, written

(9.31)

In general for a matrix

(9.32)

The three-dimensional analog of a parallelogram is a parallelepiped, with all six faces being parallelograms. As shown in Figure 9.2, the parallelepiped is oriented between the origin and the point , which is the reflection of the origin through the plane containing the points , and . You can figure out, using some algebra and trigonometry, that the volume is given by

(9.33)

[Using vector analysis, , where are the vectors from the origin to , respectively.]

It might be conjectured that an -determinant represents the hypervolume of an -dimensional hyperparallelepiped.

In general, a determinant is given by

(9.34)

A determinant can be evaluated by summing over products of elements along the two diagonals, northwest-southeast minus northeast-southwest:

Similarly for a determinant:

where the first two columns are duplicated on the right. There is no simple graphical method for or larger determinants. An determinant is defined more generally by

(9.35)

where is a permutation operator which runs over all possible permutations of the indices The permutation label is even or odd, depending on the number of binary interchanges of the second indices necessary to obtain , starting from its order on the main diagonal: . Many math books show further reductions of determinants involving minors and cofactors, but this is no longer necessary with readily available computer programs to evaluate determinants. An important property of determinants, which is easy to verify in the and cases, is that if any two rows or columns of a determinant are interchanged, the value of the determinant is multiplied by . As a corollary, if any two rows or two columns are identical, the determinant equals zero.

The determinant of a product of two matrices, in either order, equals the product of their determinants. More generally for a product of three or more matrices, in any cyclic order,

(9.36)

9.4 Matrix Inverse

The inverse of a matrix , designated , satisfies the matrix equation

(9.37)

For the matrix

The inverse is given by

(9.38)

For matrices of larger dimension, the inverses can be readily evaluated by computer programs. Note that the denominator in (9.38) equals the determinant of the matrix . In order for the inverse to exist, the determinant of a matrix must not be equal to zero. Consequently, a matrix with determinant equal to zero is termed singular. A matrix with is called unimodular.

The inverse of a product of matrices equals the product of inverses in reversed order. For example,

(9.39)

You can easily prove this by multiplying by .

The inverse matrix can be used to solve a series of simultaneous linear equations, such as (9.4). Supposing the are known quantities while the are unknowns, multiply the matrix equation (9.5) by . This gives

(9.40)

With the elements of and known, the column vector , hence its elements can be determined. The solutions are given explicitly by Cramer’s rule:

(9.41)

where is the determinant of the matrix :

(9.42)

and is obtained from by replacing the ith column by the column vector :

(9.43)

A set of homogeneous linear equations

(9.44)

always has the trivial solution. A necessary condition for a nontrivial solution to exist is that . (This is not a sufficient condition, however. The trivial solution might still be the only one.)

9.5 Wronskian Determinant

A set of functions is said to be linearly independent if vanishing of the linear combination

(9.45)

can only be achieved with the “trivial” solution

A criterion for linear independence can be obtained by constructing a set of simultaneous equations involving (9.45) along with its 1st, 2nd, …, (n − 1)st derivatives:

(9.46)

A trivial solution, hence linear independence, is guaranteed if the Wronskian determinant is nonvanishing, i.e.

(9.47)

You can show, for example, that the set is linearly independent, while the set is not.

9.6 Special Matrices

The transpose of a matrix, designated or , is obtained by interchanging its rows and columns or, alternatively, by reflecting all the matrix elements through the main diagonal:

(9.48)

A matrix equal to its transpose, , is called symmetric. Two examples of symmetric matrices are

(9.49)

If , the matrix is skew-symmetric, for example

(9.50)

A matrix is orthogonal if its transpose equals its inverse: . A unimodular orthogonal matrix—also known as a special orthogonal matrix—can be expressed in the form

(9.51)

The totality of such two-dimensional matrices is known as the special orthogonal group, designated SO(2). The rotation of a Cartesian coordinate system in a plane, such that

(9.52)

can be compactly represented by the matrix equation

(9.53)

Since is orthogonal, , which leads to the invariance relation

(9.54)

As a general principle, a linear transformation preserves length if and only if its matrix is orthogonal.

The Hermitian conjugate of a matrix, , is obtained by transposition accompanied by complex conjugation:

(9.55)

A matrix is Hermitian or self-adjoint if . The matrices , and introduced above are all Hermitian. The Hermitian conjugate of a product equals the product of conjugates in reverse order:

(9.56)

analogous to the inverse of a product. The same ordering is true for the transpose of a product. Also, it should be clear that a second Hermitian conjugation returns a matrix to its original form:

(9.57)

The analogous effect of double application is also true for the inverse and the transpose. A matrix is unitary if its Hermitian conjugate equals its inverse: . The set of unimodular unitary matrices constitutes the special unitary group SU(2). Such matrices can be parametrized by

(9.58)

or by

(9.59)

The SU(2) matrix group is of significance in the physics of spin- particles.

9.7 Similarity Transformations

A matrix is said to undergo a similarity transformation to if

(9.60)

where the transformation matrix is nonsingular. (The transformation is alternatively written .) When the matrix is orthogonal, we have an orthogonal transformation: . When the transformation matrix is unitary, we have a unitary transformation: . All similarity transformations preserve the form of matrix equations. Suppose

Premultiplying by and postmultiplying by , we have

Inserting in the form of between and :

From the definition of primed matrices in Eq. (9.60), we conclude

(9.61)

This is what we mean by the form of a matrix relation being preserved under a similarity transformation. The determinant of a matrix is also invariant under a similarity transformation, since

(9.62)

9.8 Matrix Eigenvalue Problems

One important application of similarity transformations is to reduce a matrix to diagonal form. This is particularly relevant in quantum mechanics, when the matrix is Hermitian and the transformation unitary. Consider the relation

(9.63)

where is a diagonal matrix, such as (9.21). Premultiplying by , this becomes

(9.64)

Expressed in terms of matrix elements:

(9.65)

recalling that the elements of the diagonal matrix are given by and noting that only the term with will survive the summation over . The unitary matrix can be pictured as composed of an array of column vectors , such that , like this:

(9.66)

Accordingly Eq. (9.64) can be written as a set of equations

(9.67)

This is an instance of an eigenvalue equation. In general, a matrix operating on a vector will produce another vector , as shown in Eq. (9.5). For certain very special vectors , the matrix multiplication miraculously reproduces the original vector multiplied by a constant , so that

(9.68)

Eigenvalue problems are most frequently encountered in quantum mechanics. The differential equation for the particle-in-a-box, treated in Section 8.6, represents another type of eigenvalue problem. There, the boundary conditions restricted the allowed energy values to the discrete set , enumerated in Eq. (8.111). These are consequently called energy eigenvalues.

The eigenvalues of a Hermitian matrix are real numbers. This follows by taking the Hermitian conjugate of Eq. (9.63):

(9.69)

Since , by its Hermitian property, we conclude that

(9.70)

Hermitian eigenvalues often represent physically observable quantities, consistent with their values being real numbers.

The eigenvalues and eigenvectors can be found by solving the set of simultaneous linear equations represented by (9.67):

(9.71)

This reduces to a set of homogeneous equations:

(9.72)

A necessary condition for a nontrivial solution is the vanishing of the determinant:

(9.73)

this is known as the secular equation and can be solved for roots .It is a general result that the eigenvectors of two unequal eigenvalues are orthogonal. To prove this, consider two different eigensolutions of a matrix :

(9.74)

Now, take the Hermitian conjugate of the equation, recalling that is Hermitian () and is real (). Thus

(9.75)

Now postmultiply the last equation by , premultiply the equation by , and subtract the two. The result is

(9.76)

If , then and are orthogonal:

(9.77)

When , although , the proof fails. The two eigenvectors and are said to be degenerate. It is still possible to find a linear combination of and so that the orthogonality relation Eq. (9.77) still applies. If, in addition, all the eigenvectors are normalized, meaning that

(9.78)

then the set of eigenvectors constitutes an orthonormal set satisfying the compact relation

(9.79)

analogous to the relation for orthonormalized eigenfunctions.

In quantum mechanics there is a very fundamental connection between matrices and integrals involving operators and their eigenfunctions. A matrix we denote as is defined such that its matrix elements correspond to integrals over an operator and its eigenfunctions , constructed as follows:

(9.80)

The two original formulations of quantum mechanics were Heisenberg’s matrix mechanics (1925), based on representation of observables by noncommuting matrices and Schrödinger’s wave mechanics (1926), based on operators and differential equations. It was deduced soon afterward by Schrödinger and by Dirac that the two formulations were equivalent representations of the same underlying physical theory, a key connection being the equivalence between matrices and operators demonstrated above.

9.9 Diagonalization of Matrices

A matrix is diagonalizable if there exists a similarity transformation of the form

(9.81)

All Hermitian, symmetric, unitary, and orthogonal matrices are diagonalizable, as is any -matrix whose eigenvalues are distinct. The process of diagonalization is essentially equivalent to determination of the eigenvalues of a matrix, which are given by the diagonal elements .

The trace of a matrix is defined as the sum of its diagonal elements:

(9.82)

This can be shown to be equal to the sum of its eigenvalues. Since

(9.83)

we can write

(9.84)

noting that . Therefore

(9.85)

9.10 Four-Vectors and Minkowski Spacetime

Suppose that at a light flashes at the origin, creating a spherical wave propagating outward at the speed of light . The locus of the wavefront will be given by

(9.86)

According to Einstein’s Special Theory of Relativity, the wave will retain its spherical appearance to every observer, even one moving at a significant fraction of the speed of light. This can be expressed mathematically as the invariance of the differential element

(9.87)

known as the spacetime interval. Equation (9.87) has a form suggestive of Pythagoras’ theorem in four dimensions. It was fashionable in the early years of the 20th century to define an imaginary time variable , which together with the space variables , and forms a pseudo-Euclidean four-dimensional space with interval given by

(9.88)

This contrived Euclidean geometry doesn’t change the reality that time is fundamentally very different from a spatial variable. It is current practice to accept the differing signs in the spacetime interval and define a real time variable , in terms of which

(9.89)

The corresponding geometrical structure is known as Minkowski spacetime. The form we have written, described as having the signature, is preferred by elementary-particle physicists. People working in General Relativity write instead , with signature .

The spacetime variables are the components of a Minkowski four-vector, which can be thought of as a column vector

(9.90)

with its differential analog

(9.91)

Specifically, these are contravariant four-vectors, with their component labels written as superscripts. The spacetime interval (9.89) can be represented as a scalar product if we define associated covariant four-vectors as the row matrices

(9.92)

with the component indices written as subscripts. A matrix product can then be written:

(9.93)

This accords with (9.89) provided that the covariant components are given by

(9.94)

It is convenient to introduce the Einstein summation convention for products of covariant and contravariant vectors, whereby

(9.95)

Any term containing the same Greek covariant and contravariant indices is understood to be summed over that index. This applies even to tensors, objects with multiple indices. For example, a valid tensor equation might read

(9.96)

The equation applies for all values of the indices which are not summed over. The index summed from 0 to 3 is said to be contracted. Usually, the summation convention for Latin indices implies a sum just from 1 to 3, for example

(9.97)

A four-dimensional scalar product can alternatively be written

(9.98)

Covariant and contravariant vectors can be interconverted with use of the metric tensor, given by

(9.99)

For example,

(9.100)

The spacetime interval takes the form

(9.101)

In General Relativity, the metric tensor is determined by the curvature of spacetime and the interval generalizes to

(9.102)

where might have some nonvanishing off-diagonal elements. In flat spacetime (in the absence of curvature), this reduces to Special Relativity with .

The energy and momentum of a particle in relativistic mechanics can also be represented as components of a four-vector with

(9.103)

and correspondingly

(9.104)

The scalar product is an invariant quantity

(9.105)

where is the rest mass of the particle. Written out explicitly, this gives the relativistic energy-momentum relation:

(9.106)

In the special case of a particle at rest , we obtain Einstein’s famous mass-energy equation . The alternative root is now understood to pertain to the corresponding antiparticle. For a particle with zero rest mass, such as the photon, we obtain . Recalling that , this last four-vector relation is consistent with both the Planck and de Broglie formulas: and .