Matrices

• Type of physical science: Mathematical methods
• Field of study: Algebra

Many problems involving a physical system give rise to a system of equations that can be conveniently described using matrices. By analyzing properties of the resulting matrices or performing certain operations on the matrices, information about the physical system can often be obtained.

Overview

A matrix is a rectangular array of numbers. The numbers within a matrix are called entries or elements of the matrix. A matrix is named using a capital letter (B, for example), but sometimes parentheses are used instead of brackets. Each horizontal list of numbers in a matrix is called a row of the matrix, and each vertical list is called a column.

The dimension of a matrix tells the number of rows and columns it has and is written in the form (number of rows) x (number of columns). Note that the multiplication sign is read “by” and the number of rows is always written first. Mathematicians refer to an individual entry of a matrix by using a doubly subscripted variable. The variable is the lowercase form of a matrix name, and the subscript is the row number followed by the column number of the entry (called the address of the entry).

The notation for describing individual entries is particularly useful when defining matrix operations. Matrices A and B having the same dimensions are equal if each pair of entries with the same address are equal. So, A = B if a_ij = b_ij for all pairs i, j. Addition of matrices is easily described.

Subtraction is defined similarly; that is, the i, j entry of A - B is a_ij - b_ij. Scalar multiplication consists of the multiplication of a number (scalar) times a matrix. Multiplying scalar k times matrix, A = [a_ij] yields kA = [ka_ij]. Thus, each entry in A is multiplied by k.

An important class of matrices that occur in applications are those with only a single row or column. A 1 x m matrix is called a row vector, and an n x 1 matrix is a column vector. The dot product (inner product) is an operation defined on two vectors that have the same number of entries and is the sum of the products of corresponding entries. Thus, [2 4 -7 3] · [5 9 6 8] = 2(5)+4(9)+(−7)(6)+3(8) = 28. Note that the dot product of two vectors is a number, not a vector.

Matrix multiplication consists of multiplying one matrix by another. For matrices A and B, the i,j entry of the matrix product AB is the dot product of row i of A with column j of B. Therefore, in order to multiply two matrices, the number of columns of the first matrix must equal the number of rows of the second. Matrix multiplication is not commutative.

A square matrix has the same number of rows as columns. Its diagonal consists of those entries whose row number equals its column number. Thus, the product found is a square matrix and its diagonal consists of the entries 20 and 33. The identity matrix I_n is the n x n matrix with each diagonal entry equal to 1 and each other entry equal to 0.

For any square matrix A, a number called the determinant of A and denoted det(A) is associated. It is found by adding all n! signed products ±a_1ja_2ja_3j…a_nj where j₁,j₂,…j_n is a permutation of the column numbers 1, 2, 3, . . . , n, and the sign is + precisely when the permutation has an even number of pairs in “unnatural” order. For example, if then det(A) = a₁₁a₂₂a₃₃ - a₁₁a₂₃a₃₂ + a₁₂a₂₃a₃₁ - a₁₂a₂₁a₃₃ + a₁₃a₂₁a₃₂ - a₁₃a₂₂a₃₁ = 5(0) - 5 + 2(-1) - 2 + 4 - 4(0)(-1) = 0 - 10 - 2 - 36 + 24 + 0 = -24

Since finding all signed products is cumbersome, techniques for finding determinants often perform some simplifications first using row operations, including Gaussian elimination, to reduce the matrix to a simpler form. Most linear algebra texts discuss such techniques, and finite mathematics texts give a more elementary treatment of the same topic.

For any n x n matrix A with nonzero determinant, there is a unique matrix A⁻¹, the (multiplicative) inverse of A, such that AA⁻¹ = A⁻¹A = Iₙ.

Many applications give rise to a system of n linear equations in n variables x₁, x₂, …, x_n. Let A = [a_ij] be the n x n matrix of coefficients of this system, let x = [x_j be the n x 1 column vector of variables, and let c = [c_i] be the n x 1 column vector of constants. Then the system can be written in matrix form as Ax = c. It can be shown that the system has a unique solution if and only if detA ≠ 0, and in that case, the solution is x = A^-1c.

In analyzing various physical systems, one must find values of some variable, λ, such that A - λ I has no inverse where A is a square matrix arising from the attributes of the system. A - λ I has no inverse if and only if det(A - λ I) = 0. A value of λ that makes det(A - λI) = 0 is called an eigenvalue of A.

Set this equal to zero and solve to find two roots, 2 and -3. Thus, A has two eigenvalues λ = 2 and λ = -3.

Applications

Vectors are the most common matrices occurring in physics. They are used to describe displacement, velocity, acceleration, force, and other quantities that have both magnitude and direction. For example, suppose that one end of a metal bar, 70 centimeters long, is heated to 200 degrees Celsius for sixty seconds. Heat will be conducted along the bar in such a way that regions closest to the heated end are hotter than those further along the bar. The temperature distribution along the bar, measured at 10-centimeter intervals from the heated end, would be given by a vector t = [t₁​ t₂​ t₃​ … t7​].

Another instance of vectors occurs in mechanics where one wants to determine the (effective) resultant force of several forces applied to a single object. For simplicity, think of the object as a small metal ring with several ropes attached. In a “tug of war,” people are pulling on the ends of the ropes in order to get the ring to move in their direction. The resultant force describes the magnitude and direction of the combination of all the forces. That is, it corresponds to one single force that would have the same effect on the motion of the ring as the combined forces have.

Vectors are represented graphically as little arrows in a coordinate system. In the example, the origin of the coordinate system gives the original location of the metal ring. For each rope, an arrow emanates from the origin. Its direction indicates the direction of pull and its length indicates the magnitude of the pulling force. The force can then be represented by a vector whose coordinates indicate the endpoint of the arrow. The resultant force is the sum of all the (force) vectors.

In a related problem, one wants to know the force needed to put the system in equilibrium—that is, what additional force would cause the ring to remain motionless. The equilibrium force is the negative of the resultant of the other forces. For example, if forces [3 5], [-2 1], and [4-2] are applied to the ring, then the resultant force is [3 5]+[−2 1]+[4 −2]=[3+(−2)+4, 5+1+(−2)]=[5, 4]. The equilibrium vector is −[5 4]=[−5 −4].

The magnitude of the force can be found using the Pythagorean theorem. Thus, the magnitude of the resultant force [x y] = [5 4] is √(5² + 4²) = √41 ≈ 6.403.

From trigonometry, one finds that the direction of the force corresponds to the angle whose tangent is y/x = 4/5=0.8. The appropriate angle is approximately 38.7 degrees south of east.

The same ideas extend to three-dimensional space, where a force is represented by a vector [x y z]. The Pythagorean theorem in that case gives the magnitude of a vector [12-34] as √[12² + (-3)² + 4²] = √(144 + 9 + 16) = √169 = 13

Matrices commonly occur in problems dealing with oscillatory motion, such as problems involving springs, suspension bridges, or wings of an airplane in flight. These problems yield systems of differential equations. The derivatives occur because of the formula F = ma and the fact that acceleration is the second derivative of displacement.

Solving the oscillation problems involves finding eigenvalues of the matrix arising from the system of equations.

A variety of problems involving electrical networks are solved using matrices. For example, Gustav Robert Kirchhoff modeled electrical networks using graphs (of graph theory—also used to model molecules, transportation networks, and computer systems). A graph, in turn, can be modeled by a matrix A. Kirchhoff showed that the number of “spanning trees” in the graph equals the absolute value of the determinant of the matrix obtained by deleting any one row and any one column from A.

Several Nobel laureates in Physics used matrices extensively in their work. Werner Heisenberg (Nobel laureate, 1932) used matrices to help describe energy levels and electron orbits in molecules. Wolfgang Pauli (Nobel laureate, 1945) used certain 2 x 2 matrices (called Pauli spin matrices) in quantum mechanics to help represent the spin of electrons. Paul Adrian Maurice Dirac (Nobel laureate, 1933) used Pauli spin matrices to generate a set of 4 x 4 matrices that he used to show the invariance of a certain wave equation in his famous paper on the quantum theory of the electron.

Context

Although methods for solving systems of linear equations date back to the Han dynasty (206 BCE to CE 220), they led first to determinants in the late 1600s, rather than to matrices.

Around 1750, Gabriel Cramer developed his method, called Cramer’s rule, that used determinants to solve systems of equations. Matrices were developed in the mid-1800s by James Joseph Sylvester to facilitate discussion and examination of determinants. Several years later, Arthur Cayley noted that any linear transformation can be represented by a matrix. This led to much work on linear transformations and matrices.

Suppose, for example, that a uniform flat metal plate is rotated around its center of mass. Think of the center of mass as the origin of the Cartesian coordinate system. Then the location of each point on the plate can be described by a pair of coordinates (x, y).

By rotating the object 15 degrees counterclockwise, each point (x, y) is rotated into some other point (x′, y′) whose coordinates are related to those of (x, y) by: x′ = x cos(15°) - y sin(15°); y′ = x sin(15°) + y cos(15°).

A three-dimensional object, such as a building, may cast a shadow on a flat surface. The relative size and shape of the shadow depend on the angle of inclination of the sun. The shadow is a projection of a three-dimensional object (the building) onto a two-dimensional object (the flat surface), and the linear transformation (the projection) can be described by a matrix.

Around 1900, Giuseppe Peano formally defined the concept of a vector space. Throughout the twentieth century, much work was done on both the theory and applications of matrices and vector spaces. Two of the many important applications of matrices are linear programming and Markov chains.

Matrices play an important role in computer science because one of the most common methods of storing data in a computer is by using matrices. Indeed, common spreadsheet programs are all matrix-based. Matrices also play a key role in computer graphics, where the location of an image on a computer screen can be represented by a matrix describing which pixels are highlighted on the screen. Rotating or linearly expanding an image can be accomplished by multiplying the location matrix by an appropriate matrix.

Principal terms

DETERMINANT: a number associated with a square matrix; found by adding all n! signed products +/- a₁ja₂ja sub 3j. . .anj, where j₁, j₂, . . . , jn is a permutation of the column numbers, 1, 2, 3, . . . , and the sign is + precisely when the permutation has an even number of pairs in “unnatural” order

DIAGONAL: the set of entries aii of a matrix A, that is, the entries with identical row and column numbers

DIMENSION: the number of rows, n, and columns, m, in a matrix, generally written n x m (read “n by m”)

EIGENVALUE: for a square matrix A, this is any root λ of the equation det(A - λI) = 0, where det is the determinant function and I is the identity matrix with the same dimension as A

IDENTITY MATRIX: a square matrix whose diagonal entries are each 1 and whose other entries are each 0; the n x n identity matrix is denoted In (or simply I)

INVERSE: for an n x n matrix A with det(A) ≠ 0, this is the unique matrix A⁻¹ such that AA⁻¹ = A⁻¹A = Iₙ, where Iₙ is the n × n identity matrix

LINEAR EQUATION: an equation in which each variable has exponent 1 and of the form a₁x₁ + a₂x₂ + … + a_nx_n = k, where ai and k are numbers and the xi are variables

MATRIX: a rectangular array of numbers (called entries or elements) arranged in rows and columns

SQUARE MATRIX: a matrix with dimension n x n; that is, a matrix with the same number of rows as columns

VECTOR: a matrix having a single row (a row vector) or a single column (a column vector); often used to indicate a quantity that has both magnitude and direction, such as velocity, displacement, or force

Bibliography

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