In this chapter, we review some of the basic results from the theory of matrices and determinants.

2.1 Matrices

Let us denote by Mat _m × n the set of m × n matrices (that is, m rows and n columns) with real entries. When m = n , we say that the matrices are square. It is easily shown that with the usual matrix addition and scalar multiplication, Mat _m × n is a vector space, and that with the usual matrix multiplication, Mat _m × m is a ring.

Let P be a matrix in Mat _m × n , with

The transpose of P is the matrix P ^T in Mat _n × m defined by

The row matrices of P are

and the column matrices of P are

We say that a matrix in Mat _m × m is symmetric if Q = Q ^T , and diagonal if for all i ≠ j . Evidently, a diagonal matrix is symmetric. Given a vector (a ¹, … , a ^m ) in ℝ ^m , the corresponding diagonal matrix is defined by

where all the entries not on the (upper‐left to lower‐right) diagonal are equal to 0. For example,

The zero matrix in Mat _m × n , denoted by O _m × n , is the matrix that has all entries equal to 0. The identity matrix in Mat _m × m is defined by

so that, for example,

We say that a matrix Q in Mat _m × m is invertible if there is a matrix in Mat _m × m , denoted by Q ^–1 and called the inverse of Q, such that

It is easily shown that if the inverse of a matrix exists, then it is unique.

Multi‐index notation, introduced in Appendix A, provides a convenient way to specify submatrices of matrices. Let 1 ≤r ≤ m and 1 ≤s ≤ n be integers, and let I = (i ₁, … , i _r ) and J = (j ₁, … , j _s ) be multi‐indices in and respectively. For a matrix in Mat _m × n , we denote by

the r × s submatrix of P consisting of the overlap of rows i ₁, i ₂, … , i _r and columns j ₁, j ₂, … , j _s (in that order); that is,

When r = m , in which case (i ₁, … , i _m ) = (1, … , m), we denote

and when s = n , in which case (j ₁, … , j _n ) = (1, … , n), we denote

When r = 1, so that I = (i) for some 1 ≤ i ≤ m , we have

which is the ith row matrix of P. Similarly, when s = 1, so that J = (j) for some 1 ≤ j ≤ n , we have

which is the jth column matrix of P.

For a matrix in Mat _m × m , the trace of P is defined by

2.2 Matrix Representations

Matrices have many desirable computational properties. For this reason, when computing in vector spaces, it is often convenient to reformulate arguments in terms of matrices. We employ this device often.

Let V be a vector space, let ℋ = (h ₁, … , h _m ) be a basis for V, and let v be a vector in V, with

The matrix representation of v with respect to ℋ is denoted by [v]_ℋ and defined by

We refer to a ¹, … , a ^m as the components of v with respect to ℋ. In particular,

(2.2.1)

where 1 is in the ith position and 0s are elsewhere for i = 1, … , m .

With V and ℋ as above, let W be another vector space, and let be a basis for W. Let A : V → W be a linear map, with

(2.2.2)

so that

for j = 1, … , m . The matrix representation of A with respect to ℋ and ℱ is denoted by and defined to be the n × m matrix

(2.2.3)

As an example, consider the linear map A : ℝ² → ℝ³ given by A(x, y) = (y, 2x, 3x + 4), and let ℰ and ℱ be the standard bases for ℝ² and ℝ³ , respectively. Then

By parts (a) and (b) of Theorem 2.2.2,

Let V be a vector space, and let ℋ and ℱ be bases for V. Setting A = id _V in Theorem 2.2.4 yields

(2.2.5)

This shows that is the matrix that transforms components with respect to ℋ into components with respect to ℱ. For this reason, is called the change of basis matrix from ℋ to ℱ. Let . Then (2.2.2) and (2.2.3) specialize to

(2.2.6)

for i = 1, … , m and

(2.2.7)

2.3 Rank of Matrices

Consider the n‐dimensional vector space Mat_1 × n of row matrices and the m‐dimensional vector space Mat _m × n of column matrices. Let P be a matrix in Mat _m × n . The row rank of P is defined to be the dimension of the subspace of Mat_1 × n spanned by the rows of P:

Similarly, the column rank of P is defined to be the dimension of the subspace of Mat _m × 1 spanned by the columns of P:

To illustrate, for

we have rowrank(P) = colrank(P) = 2. As shown below, it is not a coincidence that the row rank and column rank of P are equal.

In light of Theorem 2.3.2, the common value of the row rank and column rank of P is denoted by rank (P) and called the rank of P . Thus,

(2.3.1)

2.4 Determinant of Matrices

This section presents the basic results on the determinant of matrices.

Consider the m‐dimensional vector space Mat _m × 1 of column matrices and the corresponding product vector space (Mat _m × 1) ^m . We denote by (E ₁, … , E _m ) the standard basis for Mat _m × 1 , where E _j has 1 in the jth row and 0s elsewhere for j = 1, … , m . Let

be an arbitrary function, and let σ be a permutation in S _m , the symmetric group on {1, 2, … , m}.We define a function

by

for all matrices P ₁, … , P _m in Mat _m × 1 .

A function Δ : (Mat _m × 1) ^m  → ℝ is said to be multilinear if for all matrices P ₁, … , P _m , Q in Mat _m × 1 and all real numbers c,

for i = 1, … , m . We say that Δ is alternating if for all matrices P ₁, … , P _m in Mat _m × 1 ,

for all 1 ≤ i < j ≤ m . Equivalently, Δ is alternating if τ(Δ) =  − Δ for all transpositions τ in .

A function Δ : (Mat _m × 1) ^m  → ℝ is said to be a determinant function (on Mat _m × 1 ) if it is both multilinear and alternating.

Let P be a matrix in Mat _m × m , and recall that in multi‐index notation the column matrices of P are P ₍₁₎, … , P _(m) . Setting

we henceforth view P as an m‐tuple of column matrices. In this way, the vector spaces Mat _m × m and (Mat _m × 1) ^m are identified. Accordingly, we now express the determinant function in Theorem 2.4.4 as

so that

where det (P) is referred to as the determinant of P. In particular, the condition det(E ₁, … E _m ) = 1 in Theorem 2.4.4 becomes

(2.4.2)

Let P be a matrix in Mat _m × m . The ij‐th cofactor of P is defined by

where ^ indicates that an expression is omitted. Thus, is the matrix in Mat_{(m − 1) × (m − 1)} obtained by deleting the ith row and jth column of P. The adjugate of P is the matrix in Mat _m × m defined by

The next result is not usually included in an overview of determinants, but it will prove invaluable later on.

It was remarked in connection with Theorem 2.2.3 that Mat _m × m is a ring under the usual operations of matrix addition and matrix multiplication, hence it is a group under matrix addition. But Mat _m × m is clearly not a group under matrix multiplication because not all matrices have an inverse. However, Mat _m × m contains a number of matrix groups. The largest of these is the general linear group, consisting of all invertible matrices:

where the characterization using determinants follows from Theorem 2.4.8.

We say that a matrix P in Mat _m × m is orthogonal if P ^T P = I _m . If so, then P ^T = P ⁻¹, hence P P ^T = P P ⁻¹ = I _m . Thus, P ^T P = I _m if and only if P P ^T = I _m . The orthogonal group is the subgroup of GL (m) consisting of all orthogonal matrices:

For a matrix P in O (m), we have

hence det(P) =  ± 1. The special orthogonal group is the subgroup of O (m) consisting of all matrices with determinant equal to 1:

2.5 Trace and Determinant of Linear Maps

The trace and determinant of a matrix were defined in Section 2.1 and Section 2.4, respectively. We now extend these concepts to linear maps.

Let V be a vector space, let ℋ be a basis for V, and let A : V → V be a linear map. The trace of A is defined by

(2.5.1)

and the determinant of A by

(2.5.2)