St. Franecis Institute of Technology PDF

St. Franecis Institute of Technology Department of Basic Science and Humanities(FE) R Prove that every Hernitian Matrix Acan be written as B+iC where Bis Symmetric matrix....

St. Franecis Institute of Technology Department of Basic Science and Humanities(FE) R Prove that every Hernitian Matrix Acan be written as B+iC where Bis Symmetric matrix. real 9. Express the Hermitian Matrix A = 2-i 2 2+i 3 -2i1| Symmetric and Cis Skew Symmetric matrix. 2i where Bis real 1.L2. Orthogonal Matrix: Symmetric and Cis Areal square matrix Ais Teal called orthogonal if AA' = A' A=1 Properties of Orthogonal Matrices: IfA is an Orthogonal Matrix then IfA is an Orthogonal |A|=t1 If Ais an Matrix then A' and A-1 are also Orthogonal If Aand B are twwo Matrix then orthogonal. A- exists and is equal Orthogonal square matrices of order ntothenA'. AB and BA are also orthoGo Exercise 1.2 1. Check ifthe following matrix is -8 orthogonal and hence find A- where A= 2 [-2 2 1 2 2. 1 Is the matrix xA=4 1 4 -8 orthogonal? If not, can it be converted into an o -2 2 how? a orthogonal matrix? If yes, 3. If A = 1 b|is orthogonal then find a, b, c. Also find A-1 C 1. Check if the Practice Exercise1.2 following matrices are orthogonal and hence find A-1 1 1 1 V3 COS a V3 V6 V2 (i) A= sin a] J2 -2 (ii) A = 1 0 (iii) A 1 2 0 V2 1 -/3 sin a 0 COSaJ V3 1 1 1 [-3 4 127 2. Is the matrixA =|12 -3 4 orthogonal? If not, can it be converted into an 4 12 -3! orthogonal matrix? If yes, how? 2 3 -6 3. Is the matrixA: 2 -1 orthogonal? If not, can it be converted into an orthogonal matrix? If yes, l2 5 how? [-8 4 a | is orthogonal then find a, b, c. Also find A-1 4 7 C a 5. If 3A =-2 1 2 is orthogonal then find a, b, c. Also find A-1 1 -2 2 14  Francis Institute St.Department of Technology o fBasic Science and Humanities(FE) LL3. Unitary Matri Wsouare matrix Ais saidto be lInitary if AA =|= A'A Proncrties of unitary matrices Eor aUnitary matrix A-1= 8 Determinant of aUnitary matrix is of modulus unity i.e. if |A| =a+ib then la +ib| = Va' +b =1 " IfAis unitarv,,then A-1 and A' are also unitary. Aand Bare two unitary matrices, then AB is also unitary. Exercise 1.3 2i 1. Showthat the matrix A=2 2i+i 2-is unitary and hence find 4-1 -ß + iS] Show that the matrix A = lB+ iy is a- iy is unitary if a' +B² +y² + 82 = 1 0 3. IfN=L1+2i then show that (| - N)+ N) is a unitary matrix. Practice Exercise 1.3 V2 -iV2 0 1. Showthat the matrix A=;iv2 -/2 o is unitary and hence find A-1 0 2 2. Check if the matrix A= 2l1 +i 1-il is unitary and hence find A-1 [2 +3i 1+2i -3i] 3. Check if the matrix A=1-3i 2-3i 0 is unitary i -i 2 L2. Echelon form, normal form, PAQ form and rank ofa matrixX 1.2.1. Elementary transformations Elementary row transformations Interchange of rows: RË > R; or R denotes interchange of /' row with jth row Scalar multiplication of a row: RË ’ kRË or kR; denotes multiplication of i" row with non-zero scalar k. Adding scalar multiple of a row to other row: R ’ R, + kR; or R, + kR; denotes that ith row is replaced with addition of /t" row with scalar multiplication of jth row. Elementary column transformations Interchange of columns: C; C; or C, denotes interchange of /ih column with jth column Scalar multiplication of a column: C;’ kC; or kC; denotes multiplication of /'" column with non-zero scalar k. Adding scalar multiple of acolumn to other column: C’ C;+ kC; or C + kC; denotes that i column is replaced with addition of ¡th column with scalar multiplication of jth column. 1.2.2.. Echelon form of a matrix Amatrix Ais said to be in echelon form if a. If there are rows containing all zeros, then they are at the bottom of the matrix b. First non-zero element of a row is right to the first non-zero entry of the previous row. (Alternatively number of zeros before first non-zero element of a row should be more than that of previous row) 15  ENales 0 2 3 |7 13 0 6 2 41 lo 0 ol lo 0 o lo 0 o2o4 Matres 4. BandCare nottin echelon form, but Dand Eare in echelon form matrix 1.23. Rank ofa number-of non-zero rows in echelon matrix isthe Rank of a form. Esamples: |7 1 3 J0 6 2 41 5 6 B =|olo 0o 02 4o lo 0 0! Rank of matrix Ais 2and that of B is 3 Note: Echelon form ofa given matrix is not unique; it depends upon the a matrix is unique. sequence of transformations. But rank of For a square matrix Aor order rn, if|A] :# 0then rank(A) = n Working rule to reduce a matrix to Echelon form by row and if ]A| =0 then rank(A)