Chapter 9

Matrix Algebra

Thus far, we have been doing algebra involving numbers and functions. It is also possible to apply the operations of algebra to more general types of mathematical entities. In this chapter, we will deal with matrices, which are ordered arrays of numbers or functions. For example, a matrix which we designate by the symbol image can represent a collection of quantities arrayed as follows:

image (9.1)

The subscripts image and image on the matrix elementsimage label the rows and columns, respectively. The matrix image shown above is an imagesquare matrix, with image rows and image columns. We will also make use of imagecolumn matrices or column vectors such as

image (9.2)

and imagerow matrices or row vectors such as

image (9.3)

Where do matrices come from? Suppose we have a set of image simultaneous relations, each involving image quantities image:

image (9.4)

This set of image relations can be represented symbolically by a single matrix equation

image (9.5)

where image is the image matrix (9.1), while image and image are image column vectors, such as (9.2).

9.1 Matrix Multiplication

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

image (9.6)

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

image (9.7)

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

image (9.8)

We can write the same equations in matrix notation:

image (9.9)

Evidently, image can be represented as a matrix product:

image (9.10)

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

image (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:image

The three image Pauli spin matrices

image (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

image (9.13)

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

image (9.14)


image (9.15)

Matrix multiplication remains associative, however, so that

image (9.16)

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

image (9.17)

Problem 9.1.1

Calculate and compare the matrix products image and image.

Problem 9.1.2

The commutator of two matrices is defined by


and the anticommutator by


Calculate the commutators and anticommutators for each pair of Pauli matrices.

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

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

The null matrix has all its elements equal to zero:

image (9.19)

As expected,

image (9.20)

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

image (9.21)

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

image (9.22)

A diagonal matrix is sometimes represented in a form such as

image (9.23)

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

image (9.24)


image (9.25)

As expected, for an arbitrary matrix image:

image (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 image and the other three at image, image, and image. Using Pythagoras’ theorem, the two sides image and the diagonal image have the lengths

image (9.27)

The area of the parallelogram is given by

image (9.28)

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

image (9.29)

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

image (9.30)

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

image (9.31)

In general for a image matrix image

image (9.32)


Figure 9.1 Area of parallelepiped equals determinant image.

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 image and the point image, which is the reflection of the origin through the plane containing the points image, and image. You can figure out, using some algebra and trigonometry, that the volume is given by

image (9.33)

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


Figure 9.2 Volume of parallelepiped equals determinant image.

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

In general, a image determinant is given by

image (9.34)

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


Similarly for a image determinant:

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

image (9.35)

where image is a permutation operator which runs over all image possible permutations of the indices image The permutation label image is even or odd, depending on the number of binary interchanges of the second indices necessary to obtain image, starting from its order on the main diagonal: image. 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 image and image cases, is that if any two rows or columns of a determinant are interchanged, the value of the determinant is multiplied by image. 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,

image (9.36)

Problem 9.3.1

Find the volume of a unit cube coincident with the coordinate axes by evaluating a image determinant.

Problem 9.3.2

As a more challenging variant, calculate the volume of a rotated unit cube with one vertex standing on the origin.

9.4 Matrix Inverse

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

image (9.37)

For the image matrix


The inverse is given by

image (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 image. In order for the inverse image 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 image is called unimodular.

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

image (9.39)

You can easily prove this by multiplying by image.

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

image (9.40)

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

image (9.41)

where image is the determinant of the matrix image:

image (9.42)

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

image (9.43)

A set of homogeneous linear equations

image (9.44)

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

Problem 9.4.1

Find the inverses of the three Pauli matrices, image, and image.

Problem 9.4.2

Using Cramer’s rule, solve the set of simultaneous linear equations


9.5 Wronskian Determinant

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

image (9.45)

can only be achieved with the “trivial” solution


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

image (9.46)

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

image (9.47)

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

Problem 9.5.1

Test the pair of functions image for linear independence.

Problem 9.5.2

Similarly test the set image.

9.6 Special Matrices

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

image (9.48)

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

image (9.49)

If image, the matrix is skew-symmetric, for example

image (9.50)

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

image (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

image (9.52)

can be compactly represented by the matrix equation

image (9.53)

Since image is orthogonal, image, which leads to the invariance relation

image (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, image, is obtained by transposition accompanied by complex conjugation:

image (9.55)

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

image (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:

image (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: image. The set of image unimodular unitary matrices constitutes the special unitary group SU(2). Such matrices can be parametrized by

image (9.58)

or by

image (9.59)

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

9.7 Similarity Transformations

A matrix image is said to undergo a similarity transformation to image if

image (9.60)

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


Premultiplying by image and postmultiplying by image, we have


Inserting image in the form of image between image and image:


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

image (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

image (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

image (9.63)

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

image (9.64)

Expressed in terms of matrix elements:

image (9.65)

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

image (9.66)

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

image (9.67)

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

image (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 image, 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):

image (9.69)

Since image, by its Hermitian property, we conclude that

image (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):

image (9.71)

This reduces to a set of homogeneous equations:

image (9.72)

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

image (9.73)

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

image (9.74)

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

image (9.75)

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

image (9.76)

If image, then image and image are orthogonal:

image (9.77)

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

image (9.78)

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

image (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 image is defined such that its matrix elements correspond to integrals over an operator image and its eigenfunctions image, constructed as follows:

image (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.

Problem 9.8.1

Find the eigenvalues and normalized eigenvectors for each of the three Pauli matrices, image, and image.

Problem 9.8.2

Find the eigenvalues and normalized eigenvectors of the matrix


Problem 9.8.3

Show that the matrix representation of a Hermitian operator, as defined in Eq. (8.166), corresponds to a Hermitian matrix.

9.9 Diagonalization of Matrices

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

image (9.81)

All Hermitian, symmetric, unitary, and orthogonal matrices are diagonalizable, as is any image-matrix whose image 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 image.

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

image (9.82)

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

image (9.83)

we can write

image (9.84)

noting that image. Therefore

image (9.85)

Problem 9.1.1

Find similarity transformations which diagonalize the Pauli matrices image and image.

9.10 Four-Vectors and Minkowski Spacetime

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

image (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

image (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 image, which together with the space variables image, and image forms a pseudo-Euclidean four-dimensional space with interval given by

image (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 image, in terms of which

image (9.89)

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

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

image (9.90)

with its differential analog

image (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

image (9.92)

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

image (9.93)

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

image (9.94)

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

image (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

image (9.96)

The equation applies for all values of the indices which are not summed over. The index image 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

image (9.97)

A four-dimensional scalar product can alternatively be written

image (9.98)

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

image (9.99)

For example,

image (9.100)

The spacetime interval takes the form

image (9.101)

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

image (9.102)

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

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

image (9.103)

and correspondingly

image (9.104)

The scalar product is an invariant quantity

image (9.105)

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

image (9.106)

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

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