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Matrices, Vectors & Fourier Analysis Summer 2017-2018 Matrices Matrix: A matrix is a rectangular array of numbers of the form mn m m n n a a a a a a a a a A . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 2 22 21 1 12 11 The numbers mn a a a ,..., , 12 11 are called the elements or entries of the matrix. The size of a matrix is described in terms of the number of rows (horizontal lines) and columns (vertical lines) it contains. To address the matrix size, it is called m×n or m by n size. In the size description, the 1 st number always denotes the number of rows and 2 nd denotes the number of column. Matrices are generally denoted by capital letter A ,B etc. Square brackets “[ ]” or curve brackets “( )” are used for the matrices notation. The entry that occurs in row i and column j of a matrix will be denoted by a ij . A general m×n matrix can be denoted as n m ij a ] [ or ] [ ij a . Example: Some examples of matrices are matrix 2 3 matrix 2 3 4 1 0 3 2 1 by , matrix 4 1 matrix 4 1 3 0 1 2 by , matrix 3 3 matrix 3 3 0 0 0 1 2 1 0 2 by e , matrix 1 2 matrix 1 2 3 1 by , matrix 1 1 matrix 1 1 4 by Classification of matrices: A classification of matrices is, in a broad sense, as follows: 1 Matrix Matrix Rectan gular Rectan gular Square Square Nonsingular Singular
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Matrices, Vectors & Fourier Analysis Summer 2017-2018 2
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Matrices, Vectors & Fourier Analysis Summer 2017-2018 Rectangular matrix: A matrix will be rectangular if it has m rows and n columns where m≠n . Example: [ 2 3 4 5 6 7 ] Square matrix: A matrix in which the number of rows is equal to the number of columns is called a square matrix. Thus a n m matrix will be a square matrix if m = n and it will be referred as a square matrix of order n . The following matrix is square. nn n n n n a a a a a a a a a A . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 2 22 21 1 12 11 The entries a 11 ,a 22 ,…,a nn are called the main diagonal entries of A . Singular matrix: The square matrix A is said to be singular if and only if (iff) its determinant is zero, i.e., det ( A ) = 0 . Nonsingular matrix: The square matrix A is said to be nonsingular iff det ( A ) 0 . Row and Column matrices: A row matrix is defined as a matrix having a single row and a column matrix is a matrix having a single column. Example: n a a 1 11 . . . . is a row matrix. 3
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Matrices, Vectors & Fourier Analysis Summer 2017-2018 Example: 1 21 11 . . . m a a a is a column matrix. Equal matrix: Two matrices are said to be equal if they have the same size and their corresponding entries are equal.
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