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PRELIMINARIES

A matrix is a way of organizing information.

It is a rectangular array of elements arranged in rows and columns. For example, the following matrix

A has m rows and n columns.

a11

a21

A = a31

M

a m1

a12

a13

a22 a32 M a m2

a 23 a 33

M

a m3

a1n

... a 2 n

... a3 n

O M

L amn

...

All elements can be identified by a typical element a ij , where i=1,2,…,m denotes rows and j=1,2,…,n denotes columns.

A matrix is of order (or dimension) m by n (also denoted as (m x n)).

A matrix that has a single column is called a column vector.

A matrix that has a single row is called a row vector.

TRANSPOSE

The transpose of a matrix or vector is formed by interchanging the rows and the columns. A matrix of order (m x n) becomes of order (n x m) when transposed.

For example, if a (2 x 3) matrix is defined by

a11 a12 a13

A=

a21 a22 a23

Then the transpose of A, denoted by A’, is now (3 x 2)

a11

A′ = a12

a13

•

•

a 21 a22 a 23

( A′)′ = A

(kA)′ = kA′ , where k is a scalar.

SYMMETRIC M ATRIX

When A′ = A , the matrix is called symmetric. That is, a symmetric matrix is a square matrix, in that it has the same number of rows as it has columns, and the off-diagonal elements are symmetric (i.e. a ij = a ji for all i and j ).

For example,

4 5 − 3

A = 5 7 2

− 3 2 10

A special case is the identity matrix, which has 1’s on the diagonal positions and 0’s on the offdiagonal positions.

1

0

I =

M

0

0 L 0

1 L 0

M O M

0 L 1

The identity matrix is a diagonal matrix, which can be denoted by diag ( a1 , a2 ,..., an ) , where a i is the ith element on the diagonal position and zeros occur elsewhere. So, we can write the identity matrix as

I = diag(1,1,...,1) .

ADDITION AND S UBTRACTION

Matrices can be added and subtracted as long as they are of the same dimension. The addition of matrix A and matrix B is the addition of the corresponding elements of A and B. So, C = A + B implies that cij = a ij + bij for all i and j.

For example, if

2 − 3

A=

6 10

0 6

B =

5 − 8

Then

2 3

C=

11 2

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•

•

A± B = B ± A

( A ± B) ± C = A ± (B ± C )

( A ± B)′ = A′ ± B′

2

M ULTIPLICATION

If k is a scalar and A is a matrix, then the product of k times A is called scalar multiplication. The product is k times each element of A. That is, if B = kA , then bij = kaij for all i and j.

In the case of multiplying two matrices, such as C = AB , where neither A nor B are scalars, it must be the case that the number of columns of A = the number of rows of B

So, if A is of dimension (m x p) and B of dimension (p x n), then the product, C, will be of order (m x

n) whose ijth element is defined as p cij = ∑ a ik bkj k =1

In words, the ijth element of the product matrix is found by multiplying the elements of the ith row of A, the first matrix, by the corresponding elements of the jth column of B, the second matrix, and summing the resulting product. For this to hold, the number of columns in the first matrix must equal the number of rows in the second.

For example,

6 8 3 − 8 1

F = AD =

− 2 4 9 2 5

6 * (−8) + 8 * 2

6 *1 + 8 * 5

6* 3 + 8* 9

=

( −2) * 3 + 4 * 9 ( −2) * ( −8) + 4 * 2 (−2) *1 + 4 * 5

90 − 32 46

=

30 24 18

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•

A (m x 1) column vector multiplied by a (1 x n) row vector becomes an (m x n) matrix.

A (1 x m) row vector multiplied by a (m x 1) column vector becomes a scalar.

In general, AB ≠ BA .

But, kA = Ak if k is a scalar and A is a matrix.

And, AI = IA if A is a matrix and I is the identity matrix and conformable for multiplication.

The product of a row vector and a column vector of the same dimension is called the inner product

(also called the dot product), its value is the sum of products of the components of the vectors. For example, if j is a (T x 1) vector with elements 1, then the inner product, j’j, is equal to a constant T.

Note: two vectors are orthogonal if their inner product is zero.