Introduction to Matrix Algebra - Appalachian State University Introduction to Matrix Algebra George

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Transcript of Introduction to Matrix Algebra - Appalachian State University Introduction to Matrix Algebra George

  • Introduction to Matrix Algebra

    George H Olson, Ph. D.

    Doctoral Program in Educational Leadership

    Appalachian State University

    September 2010

    What is a matrix? Dimensions and order of a matrix. A p by q dimensioned matrix is a p (rows) by q (columns) array of numbers, or symbols, and will itself be represented by an upper-case, bold, italic symbol; e.g, A , β . For instance, a 2 by 3 matrix, A , can

    be represented, as a symbolic array,

     

      

     

    23

    13

    22

    12

    21

    11

    a

    a

    a

    a

    a

    a A ,

    or as an actual array of numbers

     

      

     

    3

    6

    4

    5

    2

    3 A .

    In either case, the order of A is said to be 2  3. On the other hand, the matrix,

       

       

    434241

    333231

    232221

    131211

    ddd

    ddd

    ddd

    ddd

    D ,

    is a 4  3 matrix. That is, its order is 4  3. A matrix having only one column or one row is called a vector and is represented by a lower-case, bold, italic symbol; e.g., a, β . Hence the 1  4 and 3  1 matrices,

     4321 aaaaa , and

  •   

      

    3

    2

    1

    β ,

    are both vectors. A is a row vector; β is a column vector.

    The numbers (or symbols) inside a matrix are called elements. Thus, in the first matrix, A, given above, a11 and a23 are elements of the matrix A. Similarly, in the second matrix, A, given earlier, the numbers 4 and 6 are elements. Elements in a matrix are indexed (or referred to) by subscripts that give their row and column locations. For, instance, in the first matrix, A, above, a23 is the element in the second row of the third column. Similarly, in the matrix, D, above, element d41 is found in the fourth row, first column. In general, aij is the i,j’th element of A. When referring to elements of vectors, however, the first row (or column) is assumed. Hence, in the vector, a, above, a2 is its second element; in

    β , 3 is the third element. In general, elements of vectors, are indicated by the

    j’th element of row (i.e., 1 x q) vectors and the i’th element of column (i.e., p x 1) vectors.

    A matrix in which the number of rows (p) equals the number of columns (q) is called a square matrix.; otherwise, providing it is not a vector, it is called a rectangular matrix.

    The natural order of a rectangular matrix has more rows than columns (i.e., p > q). Hence,

      

      

    23

    22

    21

    13

    12

    11

    a

    a

    a

    a

    a

    a

    A ,

    a 3 x 2 matrix, is presented in its natural order.

    The transpose of a matrix is obtained by interchanging its rows and columns. Thus, the transpose of A, represented as A , is

     

      

     

    23

    13

    22

    12

    21

    11

    a

    a

    a

    a

    a

    a A ,

    a 3 x 2 matrix. As a concrete example, consider

  •    

       

    622

    761

    475

    342

    X ,

    which, when transposed, becomes

      

      

    

    6

    2

    2

    7

    6

    1

    4

    7

    5

    3

    4

    2

    X .

    The natural order of a vector is a p x 1 column vector. Here, the 4 x 1 column vector,

       

       

    4

    3

    2

    1

    a

    a

    a

    a

    a ,

    is presented in its natural order. The transpose of a,

     4321 aaaaa ,

    is a 1 x 4 row vector.

    As mentioned earlier a matrix where p = q is a square matrix. At least two particular types of square matrices are particularly important, symmetric matrices and diagonal matrices. A symmetric matrix is a square matrix in which the elements above the main diagonal are mirror images of the elements below the main diagonal (the main diagonal is that set of elements running from the upper left-hand side of a square matrix to the lower right-hand. In the symmetric matrix V, given below, the elements in bold represent the main diagonal. Note that the elements above the main diagonal are mirror images of the elements below the main diagonal.

      

      

    59

    64

    56

    6138

    6152

    3852

    V .

    Diagonal matrices are square matrices in which all elements except the main diagonal are zero (0). For instance,

  •    

       

    61000

    04900

    00520

    00037

    D

    is a symmetric diagonal matrix.

    An important diagonal matrix is the identity matrix. The identity matrix is a diagonal matrix in which all the elements along the main diagonal are unity (1). For example,

      

      

    100

    010

    001

    I

    is a 3 x 3 identity matrix.

    Symmetric matrices have important properties. For instance, a symmetric matrix is equal to its transpose. That is, for the matrix given a little earlier, V = V .

  • Matrix Operations

    Addition

    The addition of two matrices, A and B, is accomplished by adding corresponding

    elements, e.g., aij + bij.; hence, for the 4 x 3 matrices, andA B .

       

       

       

       

    

    

    

    

       

       

       

       

    

    434241

    333231

    232221

    131211

    434342424141

    333332323131

    232322222121

    131312121111

    434241

    333231

    232221

    131211

    434241

    333231

    232221

    131211

    ccc

    ccc

    ccc

    ccc

    bababa

    bababa

    bababa

    bababa

    bbb

    bbb

    bbb

    bbb

    aaa

    aaa

    aaa

    aaa

    BAC

    It is obvious (or should be obvious) that the orders of the two matrices being added need to be identical. Here, for instance, both matrices are of order 4 x 3. When the orders of the two matrices are not identical, addition is not impossible.

    Addition of matrices is commutative. Hence A + B = B + A.

    As a concrete example, let

       

       

    431

    854

    113

    857

    A , and

       

       

    313

    214

    331

    302

    B .

  • Then, C = A + B = B + A

    =

       

       

    

    

    

    

    341331

    281544

    313113

    380527

    =

       

       

    744

    1068

    444

    1159

    .

    Multiplication

    Multiplication is somewhat more complicated. When multiplying matrices we need to distinguish among multiplication by a scalar, vector multiplication, pre- multiplication, and post-multiplication, inner-products and outer-products. But first, let us define a scalar. A scalar is simply a single number, such as 2, 17.6, or -300.335. Symbolically, we typically use the symbol, c, to represent a scalar.

    Scalar multiplication

    Scalar multiplication is easily shown by example. Given the 3 x 2 matrix X and the scalar, c, the product, cX, is given by

    Note that all the elements in X are multiplied by the scalar, c.

    .

    3231

    22

    3231

    22

    3231

    22

    3231

    22

      

      

      

      

      

      

      

      

    cxcx

    cxc

    cc

    c

    xx

    x