Matrix Types & Operations PDF

7. Types of matrices ▪Identity matrix ▪The inverse of a matrix ▪The transpose of a matrix ▪Symmetric matrix ▪Orthogonal matrix 1  Identity matrix  a11 a1n  ▪A square matrix whose...

7. Types of matrices ▪Identity matrix ▪The inverse of a matrix ▪The transpose of a matrix ▪Symmetric matrix ▪Orthogonal matrix 1  Identity matrix  a11 a1n  ▪A square matrix whose a12  0 a22 a2 n  elements aij = 0, for i > j is called     upper triangular, i.e.,   0 0 ann    Identity matrix ▪A square matrix whose  a11 0 0  a 0  elements aij = 0, for i < j is called  21 a22  lower triangular, i.e.,      an1 an 2 ann   ▪Both upper and lower  a11 0 0   0 0  triangular, D= a22    i.e., aij = 0, for i  j , i.e.,    0 0 ann  is called a diagonal matrix, simply D = diag[a11 , a22 ,..., ann ]  In particular, a11 = a22 = … = ann = 1, The matrix I is called identity matrix. Properties: AI = IA = A 1 0 0  1 0  Examples of identity matrices:  and 0 1 0  0 1  0 0 1   Special square matrix ▪AB ≠ BA in general. However, if two square matrices A and B such that AB = BA, then A and B are said to be commute. Can you suggest two matrices that must commute with a square matrix A? Ans: A itself, the identity matrix,.. Then A and B are commute.  Special square matrix ▪If A and B such that AB = -BA, then A and B are said to be anti-commute. Then A and B are anti-commute  The inverse of a matrix ▪If matrices A and B such that AB = BA = I, then B is called the inverse of A (symbol: A-1); and A is called the inverse of B (symbol: B -1). 1 2 3 6 −2 −3 Example: A= 3 B=  −1 1 0 1 3    1 2 4    −1 0 1   Show B is the the inverse of matrix A. 1 0 0 Ans: Note that AB = BA =  0 1 0  Can you show the  0 0 1  details?  The transpose of a matrix ▪The matrix obtained by interchanging the rows and columns of a matrix A is called the transpose of A (write AT ). 1 3 Example: A= 2 4 5 6  1 4 The transpose of A is AT =  2 5   3 6  ▪For a matrix A = [aij], its transpose AT = [bij], where bij = aji.  Symmetric matrix ▪A matrix A such that AT = A is called symmetric, i.e., aji = aij for all i and j. 1 2 3 Example: A= 2 4 −5  is symmetric.  3 −5 6   ▪A + AT must be symmetric. Why?  Symmetric matrix ▪A matrix A such that AT = -A is called skew-symmetric, i.e., aji = -aij for all i and j. ▪A - AT must be skew-symmetric. Why? Note that:  Orthogonal matrix ▪A matrix A is called orthogonal if AAT = ATA = I, i.e., AT = A-1 1/ 3 1/ 6 −1/ 2   Example: prove that A = 1/ 3 −2 / 6 0    is orthogonal.  1/ 3 1/ 6 1/ 2    1/ 3 1/ 3 1/ 3    Since, A =  1/ 6 T −2 / 6 1/ 6 . Hence, AAT = ATA = I.    −1/ 2 0 1/ 2  Can you show the details?  3 − 6 2   3 2 6  49 0 0  1 0 0 1  1  1  T A A= 2 3 6 −6 3 2 = 0 49 0  = 0 1 0 7 7  49     6 2 − 3  2 6 − 3  0 0 49 0 0 1  AA T = AT A  A matrix A is orthogonal. Properties of inverse matrix ▪(AB)-1 = B -1A-1 ▪(AT)T = A and (A)T =  AT ▪(A + B)T = AT + BT ▪(AB)T = BT AT Example: Prove (AB)-1 = B-1A-1. Since (AB) (B-1A-1) = A(B B-1)A-1 = I and (B-1A-1) (AB) = B-1(A-1 A)B = I. Therefore, B-1A-1 is the inverse of matrix AB.  TYPES OF MATRICES NAME DESCRIPTION EXAMPLE Rectangular No. of rows is not equal to no. of  6 2 − 1 matrix columns − 2 0 5    Square No. of rows is equal to no. of columns  2 −1 3 − 2 0 1 matrix    1  2 4  Diagonal Non-zero element in principal 2 0 0 diagonal and zero in all other 0 4 0 matrix   positions 0  0 7  Scalar Diagonal matrix in which all the 4 0 0 elements on principal diagonal and 0 4 0 matrix   same 0  0 4   NAME DESCRIPTION EXAMPLE Row matrix A matrix with only 1 row 3 2 1 − 4 Column matrix A matrix with only I  2 column    3 Identity matrix Diagonal matrix having 1 0 each diagonal element 0 equal to one (I)  1   Zero matrix A matrix with all zero 0 0 entries   0 0  NAME DESCRIPTION EXAMPLE Upper Triangular Square matrix having all 2 5 3 the entries zero below the 0 6 matrix  4  principal diagonal  0 0 7  Lower Triangular Square matrix having all 2 0 0 the entries zero above the 5 matrix  4 0  principal diagonal  6 3 7   8. Row Operations Symbol Meaning Rij Interchange rows i and j. cRi Multiply the ith row by the nonzero constant c. cRi+Rj Multiply the ith row by c and add to the jth row. Finding A-1 Using Elementary Row Operations The procedure for finding A-1 is outlined in the following diagram: EXAMPLE (Inverse by Elementary Row Operations) Find the multiplicative inverse for SOLUTION Because I appears to the left of the vertical line, we conclude that the matrix to the right of the line is A-1 25-7-2022  9. Row echelon form, and reduced row echelon form A matrix is in row echelon form if its entries satisfy the following conditions 1. The first nonzero entry in each row is a 1 (called a leading 1). 2. Each leading 1 comes in a column to the right of the leading 1s in rows above it. 3. All rows of all 0s come at the bottom of the matrix. A matrix in row echelon form is in reduced row echelon form when every column that has a leading 1 has zeros in every position above and below its leading 1.  Example Row-Echelon Form Determine whether each matrix is in row-echelon form. If it is, determine whether the matrix is in reduced row-echelon form. A- B- C- D- E- F- cont’d Solution: The matrices in (A), (C), (D), and (F) are in row-echelon form. The matrices in (D) and (F) are in reduced row-echelon form because every column that has a leading 1 has zeros in every position above and below its leading 1. The matrix in (B) is not in row-echelon form because the row of all zeros does not occur at the bottom of the matrix. The matrix in (E) is not in row-echelon form because the first nonzero entry in Row 2 is not a leading 1. Activity Find the inverse (if exists) - 7 2 9   2 -4 - 6    3  5 2    Matrix Multiplication with Excel Use the EXCEL MMULT function to Step 3 – In the upper left-hand multiply the matrices: corner (B8) of this selected output range type the formula: Excel  1 −1 2   1 1  2 3 = MMULT(B1:D2,B4:C6). Excel A   B     Step 4 - Press CONTROL SHIFT 2 1 3  1 2 ENTER (not just enter) Step 1 – Enter matrix A into 1 A Matrix A B 1 C -1 D 2 cells B1:D2 and matrix B into 2 2 1 3 cells B4:C6. 3 4 Matrix B 1 1 5 2 3 Step 2 – Select the output 6 1 2 range (B8:C9) into which the 7 8 AB= 1 2 product will be computed. 9 7 11