Page 15 - Vector Analysis

§1.4 Matrices 11

Definition 1.40 (Identity matrix). The identity matrix of size n, denoted by In, is the n ˆ n
square matrix with ones on the main diagonal and zeros elsewhere. In other words,

In = [δij]nˆn ,

where δij is the Kronecker delta.

When the size is clear from the context, In is sometimes denoted by I.

Definition 1.41 (Transpose). Let A = [aij]mˆn be a m ˆ n matrix over scalar field F. The
transpose of A, denoted by AT, is the n ˆ m matrix given by [AT]ij = aji.

By the definition of product of matrices, we can easily derive the following two proposi-
tions.

Proposition 1.42. Let A P M(m, n; F) and B P M(n, ℓ; F). Then (AB)T = BTAT.

Proposition 1.43. Let A = [aij]mˆn be a m ˆ n matrix over scalar field F, and (¨, ¨)Fn and
(¨, ¨)Fm be the standard inner products on Fn and Fm, respectively. Then

(Ax, y)Fm = (x, ATy)Fn @ x P Fn, y P Fm .

Definition 1.44 (Rank and nullity of matrices). The rank of a matrix A, denoted by
rank(A), is the dimension of the vector space generated (or spanned) by its columns. The
nullity of a matrix A, denoted by nullity(A), is the dimension of the null space of A.

Remark 1.45. The matrix AT is often called the conjugate transpose of the matrix A.

Remark 1.46. The rank defined above is also referred to the column rank, and the row
rank of a matrix is the dimension of the vector space spanned by its rows. One should
immediately notice that the column rank of A equals the dimension of R(A) and the row
rank of A equalis the dimension of R(AT).

Theorem 1.47. Let A P M(m, n; F). Then rank(A) + nullity(A) = n.

Proof. Without loss of generality, we assume that nulltiy(A) = k ă n, and ␣v1, ¨ ¨ ¨ , vk( be
a basis of null(A). Then there exists n ´ k vectors ␣vk+1, ¨ ¨ ¨ , vn( such that ␣v1, ¨ ¨ ¨ , vn(
is a basis of Fn. We conclude the theorem by showing that ␣Avk+1, ¨ ¨ ¨ , Avn( is a basis of
R(A).