Page 794 -

Summary of Matrix Algebra 793

Types of Matrices

Square, symmetric, and diagonal matrices. A matrix is said to be square if the number of
rows equals the number of columns. A square matrix is said to be symmetric if its (i, j) ele-
ment equals its (j, i) element. A diagonal matrix is a square matrix in which all the off-
diagonal elements equal zero; that is, if the square matrix A is diagonal, then aij = 0 for i j.

Special matrices. An important matrix is the identity matrix, In, which is an n * n diago-
nal matrix with ones on the diagonal. The null matrix, 0n * m, is the n * m matrix with all
elements equal to zero.

The transpose. The transpose of a matrix switches the rows and the columns. That is, the
transpose of a matrix turns the n * m matrix A into the m * n matrix, which is denoted
by A′, where the (i, j) element of A becomes the (j, i) element of A′; said differently, the
transpose of the matrix A turns the rows of A into the columns of A′. If aij is the (i, j)
element of A, then A′ (the transpose of A) is

a11 a21 g an1
a12 a22 an2
A′ = ≥ f f g f ¥ .
f

a1m a2m g anm

The transpose of a vector is a special case of the transpose of a matrix. Thus the transpose
of a vector turns a column vector into a row vector; that is, if b is an n * 1 column vector,
then its transpose is the 1 * n row vector

b′ = 3b1 b2 g bn4.

The transpose of a row vector is a column vector.

Elements of Matrix Algebra: Addition and Multiplication

Matrix addition. Two matrices A and B that have the same dimensions (for example, that

are both n * m) can be added together. The sum of two matrices is the sum of their ele-
ments; that is, if C = A + B, then cij = aij + bij. A special case of matrix addition is vec-
tor addition: If a and b are both n * 1 column vectors, then their sum c = a + b is the
element-wise sum; that is, ci = ai + bi.

Vector and matrix multiplication. Let a and b be two n * 1 column vectors. Then the
n
product of the transpose of a (which is itself a row vector) with b is a′b = g i= 1aibi. Apply-

ing this definition with b = a yields a′a = g n 1 ai2.
i=

Similarly, the matrices A and B can be multiplied together if they are conformable—

that is, if the number of columns of A equals the number of rows of B. Specifically, suppose