20

CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES

1.3

Matrices and Matrix Operations

1.3.1

De…nitions and Notation

Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which can be performed on them, their properties and …nally their applications. De…nition 50 (matrix) 1. A matrix is a rectangular array of numbers. in which not only the value of the number is important but also its position in the array. 2. The numbers in the array are called the entries of the matrix. 3. The size of the matrix is described by the number of its rows and columns (always in this order). An m n matrix is a matrix which has m rows and n columns. 4. The elements (or the entries) of a matrix are generally enclosed in brackets, double-subscripting is used to index the elements. The …rst subscript always denote the row position, the second denotes the column position. For example 2 3 a11 a12 ::: a1n 6 a21 a22 ::: a2n 7 6 7 ::: ::: ::: 7 A = 6 (1.5) 6 ::: 7 4 ::: ::: ::: ::: 5 am1 am2 ::: amn =

[aij ] , i = 1; 2; :::; m, j = 1; 2; :::; n

(1.6)

Enclosing the general element aij in square brackets is another way of representing a matrix A . 5. When m = n , the matrix is said to be a square matrix. 6. The main diagonal in a square matrix contains the elements a11 ; a22 ; a33 ; ::: 7. A matrix is said to be upper triangular if all its entries below the main diagonal are 0. 8. A matrix is said to be lower triangular if all its entries above the main diagonal are 0. 9. If all the entries of a square matrix are zero, except those entries on the main diagonal, then we say the matrix is a diagonal matrix. 10. The n n identity matrix is an n n matrix having ones on the main diagonal, and zeroes everywhere else. It is usually denoted In . 11. We say that two matrices are equal whenever they have the same dimension, and their corresponding entries are equal.

1.3. MATRICES AND MATRIX OPERATIONS

21

De…nition 51 (row and column vectors) Vectors are special forms of matrices. 1. A row vector is a vector which has only one row. In other words, it is an 1 n matrix. 2. A column vector is a vector which has only one column. In other words, it is an m 1 matrix. Example 52 Here are some matrices 2 3 1 2 1. 4 5 2:5 5 is a 3 2 matrix. 0 1 3 2 1 2 3 2. 4 3 p5 9 5 is a square (3 3) matrix. 2 0 3 2 1 0 0 3. 4 0 1 0 5is the 3 3 identity matrix. 0 0 1 3 2 1 3 5 4 6 0 2 6 4 7 7 4. 6 4 0 0 2 1 5 is a 4 4 upper triangular matrix. 0 0 0 5 3 2 1 6 2 7 7 5. 6 4 3 5 is a column vector. It is also a 4 1 matrix. 10 6.

5

0

2

is a row vector. It is also a 1

3 matrix.

In the case of a vector, there is no need to use double subscripts. For example, instead of writing A = a11 a12 a13 a14 , we write A = a1 a2 a3 a4 . In the special case that m = n = 1, the matrix is a 1 1 matrix and may be written A = [a11 ] = [a] = a. In other words, subscripts are not needed. Since the matrix only has one entry, it is the same as a number (also called a scalar).

1.3.2

Operations on Matrices

For each operation, we give the conditions under which the operation can be performed. We then explain how the operation is performed. For the remaining of this section, unless speci…ed otherwise, we assume that A = [aij ] B = [bij ] C = [cij ]

22

CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES

Addition and Subtraction Only matrices having the same size can be added or subtracted. The resulting matrix has the same size. To add (subtract) two matrices having the same size, simply add (subtract) the corresponding entries. In other words, if C = A + B, then cij = aij + bij . Same for subtraction. Example 53 Examples of matrix addition/subtraction. 2

3 3 5 5+ 1

1 1. 4 2 0 2

1 2. 4 3 5 same

5 4 3

3 2 1 2 4 5 4 2 3 6 dimension.

2 1 = 2 1 2 3

4 6 3

5 6 3

3 1 2 5Cannot be done, the matrices do not have the 3

Scalar Multiplication This is multiplication of a matrix by a number. This operation can always be done. The result is a matrix of the same size. Simply multiply each entry of the matrix by the number. Example 54 Examples of multiplication of a matrix by a scalar. 2

1 1. 4 4 2 3 2.

a11 a21

3 2 0 4 1 5=4 8 12 10 a12 a22

=

3 0 4 5 40 a11 a21

a12 a22

Multiplication of a Row Vector by a Column Vector The row and column vector must have the same number of elements. This means that if the …rst vector has n entries (that is is a 1 n matrix), then the second vector must also have n entries (that is must be a n 1 matrix). The result is a 1

1 matrix or a scalar.

1.3. MATRICES AND MATRIX OPERATIONS

Suppose that A =

a1

a2

AB

:::

an

23 2

3 b1 6 b2 7 7 and B = 6 4 : 5. Then bn

= a1 b1 + a2 b2 + :::an bn n X = ai bi i=1

You will note that the result is a scalar. Example 55 Examples of multiplication of a row vector by a column vector. 2 3 2 6 1 7 7 1. 1 3 5 7 6 4 5 5 = [1 2 + 3 1 + 5 5 + 7 10] = 100 10 2 3 1 6 2 7 6 . This cannot be done, the vectors do not have the same 2. 1 3 5 4 7 3 5 4 number of elements. Matrix Multiplication Let us assume that A is m

p and B is q

n.

The product of A and B, denoted AB can be performed only if p = q. In other words, the number of columns of the …rst matrix, A must be the same as the number of rows of the second matrix, B. In the case p = q, then AB is a new matrix. Its size is m

n.

In summary, if we put next to each other the dimensions of the matrices we are trying to multiply, in this case m p and q n, then we see that we can do the multiplication if the inner numbers (p and q) are equal. The size is given by the outer numbers (m and n). Matrix multiplication is a little bit more complicated than the other operations. We explain it by showing how each entry of the resulting matrix is obtained. Let us assume that A = [aij ] is m p and B = [bij ] is p n. Let C = [cij ] = AB. Then, C is a m n matrix. cij is obtained by multiplying the ith row of A by the j th column of B. In other words, cij =

p X

k=1

aik bkj , i = 1; 2; :::; m, j = 1; 2; :::; n

24

CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES

Remark 56 Because of the condition on the sizes of the matrices, one can see easily that matrix multiplication will not be commutative. For example, if A is 3 4 and B is 4 5 then one can compute AB. Its size will be 3 5. However, BA cannot be computed. Even in cases when both AB and BA can be computed, they are unlikely to be the same. For example 2 32 3 2 3 1 2 3 1 1 1 14 14 14 4 2 3 4 5 4 2 2 2 5 = 4 20 20 20 5 3 4 5 3 3 3 26 26 26 but

2

Example 2 1 1. 4 2 3 2 1 2. 4 3 1 2 1 3. 4 3 1 2

1 4. 4 2 3 2

1 5. 4 2 3 AIn

1.3.3

1 4 2 3

1 2 3

32 1 1 2 54 2 3 3

2 3 4

3 2 3 6 4 5 = 4 12 5 18

3 9 12 18 24 5 27 36

57 Examples of matrix multiplication. 3 3 2 32 14 14 14 1 1 1 2 3 3 4 5 4 2 2 2 5 = 4 20 20 20 5 26 26 26 3 3 3 4 5 3 3 2 32 9 3 1 1 0 0 2 3 4 4 3 2 1 5 4 17 2 5=4 0 1 0 5 4 4 1 7 0 0 1 3 6 1 4 4 3 3 2 32 x + 2y + 3z x 2 3 2 1 5 4 y 5 = 4 3x + 2y + z 5 x + 3y + 6z z 3 6 3 2 3 1 1 1 2 3 6 2 2 2 7 7 3 4 56 4 3 3 3 5 cannot be done (why?) 4 5 4 4 4 3 3 2 32 2 3 1 0 0 1 2 3 3 4 5 4 0 1 0 5 = 4 2 3 4 5. In fact if A is m 0 0 1 3 4 5 4 5 =A

n, then

Transpose and trace of a matrix

The operations we have de…ned earlier, have analogous operations with real numbers. We now de…ne two operations which do not. We will see applications of these operations later on in the course. De…nition 58 (Transpose of a Matrix) If A is an m n matrix then the transpose of A, denoted AT is the n m matrix obtained from A by interchanging the rows and columns of A.

1.3. MATRICES AND MATRIX OPERATIONS Example 59

2

1 4 4 7

Example 60 2 Example 61

a11 4 a21 a31

3T 2 3 1 6 5 =4 2 9 3

2 5 8

a12 a22 a32

4 5 6

3T 2 a13 a11 a23 5 = 4 a12 a33 a13

2

3T 1 4 3 5 = 5

1

3

25 3 7 8 5 9

a21 a22 a23

3 a31 a32 5 a33

5

We are simply switching the rows and the columns. If A = [aij ] then AT = [aji ]. Note that in this process, the elements on the main diagonal remain unchanged. De…nition 62 (Trace of a Matrix) If A is a square matrix (n n) then the trace of A, denoted tr (A), is de…ned to be the sum of its entries on the main diagonal of A. tr (A) is not de…ned if A is not a square matrix. 1 4

Example 63 if A = 2

2 7

a11 Example 64 If A = 4 a21 a31

1.3.4

then tr (A) = 8. a12 a22 a32

3 a13 a23 5, then tr (A) = a11 + a22 + a33 . a33

Problems

Do numbers 1 - 5, 12 - 14, 19 - 23, 25 on pages 34-37