Mathematics CS126 for Computing
Matrix Algebra Matrix algebra occurs in many practical applications in computing (e.g. graphics, error detecting and error correcting codes), in mathematics and physics. An m n matrix is a rectangular array with m rows and n columns where m and n are positive integers. The matrix is enclosed by square brackets (although in some text-books round brackets are used. We use boldface capital letters to denote complete matrices1. For example: 11 4 A = 7 2 10 4 7 5 -6
The individual entries in a matrix are called the elements of the matrix. We denote the element in row i, column j of the matrix A by aij, that is we use the corresponding lower case letter with two subscripts denoting the row (i) and column (j) numbers. Thus a11 = 11, a21 = 7, a12 = 4, a23 = 5. If A is an n n matrix, we say that A is a square matrix of order n. A 1 n matrix is called a row vector of order n. An m 1 matrix is called a column vector of order m. When referring to the elements of a row vector the row number is (of course) always 1 and so is often omitted. Thus the element in column j of a row vector V is often denote as vj rather than v1j. Similarly the element in row i of a column vector U is usually denoted by ui rather than ui1 as the column number is always 1. and U = [1 2 3 4] 3 V = 4 5 is a row vector of order 4 and u1 = 1, u2 = 2, u3 = 3, u4 = 4.
is a column vector of order 3 and v1 = 3, v2 = 4, v3 = 5.
If the elements of a matrix are all integers we speak of a matrix over Z (the integers). If the elements of a matrix are all real numbers we speak of a matrix over R (the Reals). If the elements of a matrix all belong to the integers modulo n we speak of a matrix over Zn. Two matrices A and B are equal if and only if they are the same size that is each has the same number of rows and columns that is both are m n for some integers m and n and corresponding elements are equal that is aij = bij for 1 i m and 1 j n Matrix Addition and Subtraction We can add or subtract matrices A and B only if they are the same size. The element in row i column j of the sum of two matrices is the sum of the corresponding elements of the two matrices. Thus A and B are m n matrices, their sum C = A + B is also m n and cij = aij + bij
1
for 1 i m and 1 j n.
In hand-written text matrices are usually denoted by underlining the capital letter with a curly line.
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Similarly if A and B are m n matrices, their difference D = A B is also m n and dij = aij bij For example 11 4 A = 7 2 10 4 and 7 5 -6 for 1 i m and 1 j n. 3 7 B= 2 4 1 2 2 8 -8 5 3 then 2 8 11 12 C= A+B= 9 2 2 11 2 -4
and
14 3 D = A - B = 5 6 9 6
Scalar Multiplication Multiplication of a matrix A by a scalar a (a single number) produces a matrix of the same size as the original whose elements are the corresponding elements of A each multiplied by a. Thus if A is an m n matrix then B = aA is also m n and bij = aaij Example 3 1 2 If A = 6 5 4 3 6 then 3A = 18 15 9 12 for 1 i m and 1 j n. .
Matrix Multiplication Suppose A and B are m p and q n matrices respectively. Then we can form the matrix product AB if and only if p=q, that is if the number of columns of A is equal to the number of rows of B. We say A and B are conformant for the product AB. In this case C = AB is an m n matrix and the element in row i, column j of the product matrix C = AB is given by
p
cij = aik bkj = ai1b1 j + bi 2 b2 j + ai 3b3j ... + aipb pj
k =1
Note that to obtain the element in row i column j of AB we multiply corresponding elements in the ith row of A and the jth column of B and then add up these p multiples. The number of rows in the product matrix AB is the number of rows in A and the number of columns is the number of columns in B. Thus, for example, given that 3 7 5 4 2 3 4 6 7 A= and B = 2 4 3 then C = AB = 1 2 5 4 9 9 1 2 2 Note that since A is 2 3 and B is 3 3, AB is 2 3 and, for example, the element in row 1 column 2 is formed from row 1 of A and column 2 of B: 2 (7) + 3 4 + 4 2 = 6 Similarly the element in row 2 column 3 is formed from row 2 of A and column 3 of B: (1) (5) + (2) 3 + 5 2 = 9.
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Note that in the above example BA does not exist, since B is 3 3 and A is 2 3. We say A and B are not conformant for the product BA. Note also that square matrices are conformant for multiplication if and only if they have the same size. Even if A and B are square matrices of the same size it is NOT usually the case that AB = BA. For example, if 1 2 5 6 A= and B = 7 8 3 4 19 22 then C = AB = 43 50 23 34 and D = BA = 31 46
We say that in general matrix multiplication is not commutative. In a few special cases we can find square matrices A and B such that AB = BA, in this case we say A and B commute. The Zero Matrices The zero matrix Om n is the m n matrix with all its elements equal to zero Thus the 2 2 and 3 2 zero matrices are 0 0 0 0 O2 2 = O32 = 0 0 0 0 0 0 If the size of the matrix is clear from the context the subscripts are omitted and the zero matrix is represented simply by O. Let A be any m n matrix, then the following properties of the zero matrix are fairly obvious: A + O m n = Om n + A = A AOnp = Om p OqmA = Oqn
The Identity Matrices The identity matrix In is a square matrix of order n with ones on the main diagonal and zeros elsewhere. Thus the 2 2 and 3 3 identity matrices are 1 0 I2 = 0 1 1 0 0 I 3 = 0 1 0 0 0 1
If the size of the matrix is clear from the context the subscript is omitted and the identity matrix is represented simply by I. If A is any 2 3 matrix then AI3 = I 2 A = A. Thus multiplying a matrix the by identity matrix of the appropriate size leaves the matrix unchanged. For example 3 2 1 2 1 0 0 4 2 0 1 0 = 5 1 0 0 1 3 2 4 5 4 2 3 1 0 2 3 and = 0 1 1 2 5 1 2 4 5
More generally we find that if A is m n then AIn = Im A = A. In particular if A is square of order n we have AIn = I n A = A. Thus any square matrix commutes with the identity matrix of the same size.
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Powers of a Square Matrix Let A be a square matrix of order n. Then we can form the matrix product AA which is also n n; this product is denoted by A2. Similarly we can form the products: A3 = A2A, A4 = A3A, ......., Am = Am1 A for any integer m1
where we define A1 = A and A0 = In. We saw above matrix multiplication is not normally commutative and however it can be shown that any two powers of a square matrix A always commute, thus: A3 = A2A = AA2 and more generally Am = Am1 A = AAm1 37 54 and also C = A2 A = 81 118 for any integer m1. Thus any two powers of a matrix commute. For example, if 1 2 7 10 2 A= then A = 15 22 3 4 37 54 thus B = AA2 = 81 118
The Transpose of a Matrix The transpose of an m n matrix A is the n m matrix formed from A by interchanging the rows and columns of A. The transpose of A is denoted by AT (or sometimes by A'). Thus, for example, if
Suppose that A is m p and that B is p n, then A and B are conformant for the product AB which is m n . Note that BT is n p and that AT is p m and therefore BT and AT are conformant for the product BTAT which is n m. In fact the following result holds: (AB)T = For example, if 1 2 A= and 3 4 3 5 1 B= 1 8 0 1 then A = 2
T
BTAT 3 4 3 1 and B = 5 8 1 0
T T
1 2 3 5 1 5 21 1 thus AB = = 3 4 1 8 0 13 47 3 3 1 1 B A = 5 8 1 0 2
T T
5 13 and (AB) = 21 47 1 3
whereas
5 13 3 T = 21 47 = (AB) 4 1 3
Laws of Matrix Algebra In the laws that follow, a and b are scalars (single numbers) and A, B and C are matrices of an appropriate size such that the matrix sums and products indicated exist. Commutative Law for Addition A+B= B+A 4
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NO commutative Law for Multiplication Associative Law for Addition Associative Laws for Scalar Multiplication Associative Law for Matrix Multiplication Distributive Laws (Matrix) (Scalar) Properties of Zero Matrix Properties of the Identity Matrix Scalar Multiplication Properties
AB BA (A + B) + C = A + (B + C) a(AB) = (aA)B = A(aB) (ab)A = a(bA) (AB)C = A(BC) (A + B)C = A(B + C) = a(A + B) = (a + b)C = AC + BC AB + AC aA + aB aC + bC
O + A= A+ O = A OA = O AO = O IA = A 1A = A AI = A 0A = O
Symmetric and Anti-Symmetric Matrices If A is a square matrix with A = AT, we say that the matrix A is symmetric. If A is a square matrix with A = AT, we say that the matrix A is anti-symmetric (or skewsymmetric). The symmetric part of a square matrix A is the matrix (A + AT). The anti- (or skew-) symmetric part of a square matrix A is the matrix (A AT). Not surprisingly the symmetric part of a square matrix is symmetric! This follows since (A + A T)T = (AT + (A T)T) = (AT + A) = (A + A T)
Similarly the anti-symmetric part of a square matrix is anti-symmetric! The proof is left as an exercise. For example, if 1 2 A= 3 4 then 1 1 2.5 (A + AT ) = 2 2.5 4 and 1 0 -0.5 (A A T ) = 0 2 0.5
Matrices in Ada There is no direct support for matrix algebra in Ada. However it is relatively easy to...