Matrices Lecture PDF

1. define what a matrix is 2. identify special types of matrices, and 3. identify when two matrices are equal.  Matrices are everywhere. If you have used a spreadsheet such as Excel or Lotus or written a table, you have used a matrix. Matrices make presentation of numbers clearer and mak...

1. define what a matrix is 2. identify special types of matrices, and 3. identify when two matrices are equal.  Matrices are everywhere. If you have used a spreadsheet such as Excel or Lotus or written a table, you have used a matrix. Matrices make presentation of numbers clearer and make calculations easier to program. Look at the matrix below about the sale of tires in a Blowoutr’us store – given by quarter and make of tires. Q1 Q2 Q3 Q4 Tirestone 25 20 3 2  Michigan  5 10 15 25    Copper  6 16 7 27  If one wants to know how many Copper tires were sold in Quarter 4, we go along the row Copper and column Q4 and find that it is 27. A matrix is a rectangular array of elements. The elements can be symbolic expressions or numbers. Matrix [A] is denoted by  a11 a12....... a1n  a a 22....... a 2 n  [ A] =  21       a m1 am2....... a m n  Row i of [A] has n elements and is ai1 ai 2....ain  and column j of [A] has m elements and is  a1 j  a   2j        a m j  Each matrix has rows and columns and this defines the size of the matrix. If a matrix [A] has m rows and n columns, the size of the matrix is denoted by m×n. The matrix [A] may also be denoted by [A]mxn to show that [A] is a matrix with m rows and n columns. Each entry in the matrix is called the entry or element of the matrix and is denoted by aij where I is the row number and j is the column number of the element. The matrix for the tire sales example could be denoted by the matrix [A] as 25 20 3 2  [ A] =  5 10 15 25   6 16 7 27  There are 3 rows and 4 columns, so the size of the matrix is 3×4. In the above [A] matrix, a34 =27.  Row Vector  Diagonal Matrix  Column Vector  Identity Matrix  Submatrix  Zero Matrix  Square Matrix  Tri-diagonal  Upper Triangular Matrices Matrix  Diagonally  Lower Triangular Dominant Matrix Matrix What is a vector? A vector is a matrix that has only one row or one column. There are two types of vectors – row vectors and column vectors. Row Vector: If a matrix [B] has one row, it is called a row vector [ B] = [b1 b2 bn ] and n is the dimension of the row vector. Column vector: If a matrix [C] has one column, it is called a column vector  c1   [C ] =      c m  and m is the dimension of the vector. Example 1 An example of a row vector is as follows, [ B] = [25 20 3 2 0] [B] is an example of a row vector of dimension 5. Example 2 An example of a column vector is as follows, 25  [C ] =  5   6  [C] is an example of a row vector of dimension 5. If some row(s) or/and column(s) of a matrix [A] are deleted (no rows or columns may be deleted), the remaining matrix is called a submatrix of [A]. Example 3 Find some of the submatrices of the matrix  4 6 2 [ A] =    3 − 1 2  If the number of rows m a matrix is equal to the number of columns n of a matrix [A], (m=n), then [A] is called a square matrix. The entries a11,a22,…, ann are called the diagonal elements of a square matrix. Sometimes the diagonal of the matrix is also called the principal or main of the matrix. Give an example of a square matrix. 25 20 3  [ A] =  5 10 15  6 15 7  is a square matrix as it has the same number of rows and columns, that is, 3. The diagonal elements of [A] are a11 = 25, a 22 = 10, a33 = 7. A m×n matrix for which aij = 0, i  j is called an upper triangular matrix. That is, all the elements below the diagonal entries are zero. Example 5 Give an example of an upper triangular matrix. 10 −7 0  [ A] =  0 − 0.001 6   0 0 15005  is an upper triangular matrix. A m×n matrix for which aij = 0, j  i is called an lower triangular matrix. That is, all the elements above the diagonal entries are zero. Example 6 Give an example of a lower triangular matrix. 1 0 0 [ A] = 0.3 1 0 0.6 2.5 1 is a lower triangular matrix. A square matrix with all non-diagonal elements equal to zero is called a diagonal matrix, that is, only the diagonal entries of the square matrix can be non-zero, (aij = 0, i  j ). An example of a diagonal matrix. 3 0 0 [ A] = 0 2.1 0 0 0 0 Any or all the diagonal entries of a diagonal matrix can be zero. 3 0 0 [ A] = 0 2.1 0 0 0 0 is also a diagonal matrix. A diagonal matrix with all diagonal elements equal to one is called an identity matrix, (aij = 0, i  j and aii = 1 for all i). An example of an identity matrix is, 1 0 0 0 0 1 0 0 [ A] =  0 0 1 0   0 0 0 1 A matrix whose all entries are zero is called a zero matrix, ( aij = 0for all i and j). Some examples of zero matrices are, 0 0 0  [ A] = 0 0 0 0 0 0  [B] =   0 0 0  0 0 0  A tridiagonal matrix is a square matrix in which all elements not on the following are zero - the major diagonal, the diagonal above the major diagonal, and the diagonal below the major diagonal. An example of a tridiagonal matrix is, 2 4 0 0 2 3 9 0 [ A] =  0 0 5 2   0 0 3 6 Do non-square matrices have diagonal entries? Yes, for a m×n matrix [A], the diagonal entries are a11 , a22..., ak −1,k −1 , akk where k=min{m,n}. What are the diagonal entries of 3.2 5  6 7  [ A] =   2.9 3.2    5.6 7.8 The diagonal elements of [A] are a11 = 3.2 and a 22 = 7. A n×n square matrix [A] is a diagonally dominant matrix if n aii   | aij | for all i = 1,2,.....,n and j =1 i j n aii   | aij | for at least one i, j =1 i j that is, for each row, the absolute value of the diagonal element is greater than or equal to the sum of the absolute values of the rest of the elements of that row, and that the inequality is strictly greater than for at least one row. Diagonally dominant matrices are important in ensuring convergence in iterative schemes of solving simultaneous linear equations. Give examples of diagonally dominant matrices and not diagonally dominant matrices. 15 6 7  [ A] =  2 − 4 − 2  3 2 6  is a diagonally dominant matrix as a11 = 15 = 15  a12 + a13 = 6 + 7 = 13 a 22 = − 4 = 4  a 21 + a 23 = 2 + − 2 = 4 a33 = 6 = 6  a31 + a32 = 3 + 2 = 5 and for at least one row, that is Rows 1 and 3 in this case, the inequality is a strictly greater than inequality.  − 15 6 9  [B] =  2 − 4 2   3 − 2 5.001 is a diagonally dominant matrix as b11 = − 15 = 15  b12 + b13 = 6 + 9 = 15 b22 = − 4 = 4  b21 + b23 = 2 + 2 = 4 b33 = 5.001 = 5.001  b31 + b32 = 3 + − 2 = 5 The inequalities are satisfied for all rows and it is satisfied strictly greater than for at least one row (in this case it is Row 3).   25 5 1 C  =  64 8 1 144 12 1 is not diagonally dominant as c22 = 8 = 8  c21 + c23 = 64 + 1 = 65 When are two matrices considered to be equal? Two matrices [A] and [B] is the same (number of rows and columns are same for [A] and [B]) and aij=bij for all i and j. What would make  2 3 [A] =    6 7  to be equal to b11 3  [ B] =    6 b22  The two matrices [A] and [B] would be equal if b11=2 and b22=7. Matrix Vector Submatrix Square matrix Equal matrices Zero matrix Identity matrix Diagonal matrix Upper triangular matrix Lower triangular matrix Tri-diagonal matrix Diagonally dominant matrix