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Uploaded byAditya Vaishampayan
system linear equations and matrices
The document provides a comprehensive overview of matrices and systems of linear equations, detailing various types of matrices such as row, column, square, and diagonal matrices, along with their properties and operations. It discusses methods for solving systems of linear equations, including Gaussian elimination and Gauss-Jordan elimination, as well as the concepts of equivalent matrices, row echelon forms, and reduced row echelon forms. Furthermore, it explains the conditions for singular and non-singular matrices and lays out the framework for representing linear equations in matrix form.
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system linear equations and matrices
- 1. SYSTEM OF LINEAR EQUATIONSAND MATRICES Vector Calculus and Linear Algebra
- 2. Group Members: 140280117001ADITYA VAISHAMPAYAN 140280117002 NISHANT AHIR 140280117004 BALDANIYA RAVI KUMAR 140280117005 BHAVSAR MEET D 140280117007 MARGESH DARJI 140280117008 DARJI ROHAN RAKESH 140280117009 DESAI HARSH DINESHBHAI 140280117010 DHRUVAL NALIN SHAH 140280117011 DODIYA VEDANG T 140280117012 GHOGHARI NITIN H
- 3. Matrix : Arectangular array (arrangement) of mn numbers (real or complex) in m rows and n columns is called a matrix of order m by n written as m×n. A m×n matrix is usually written as 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 ⋯ ⋯ ⋯ 𝑎1𝑛 𝑎2𝑛 𝑎3𝑛 ⋮ ⋮ ⋮ ⋮ 𝑎 𝑚1 𝑎 𝑚2 𝑎 𝑚3 ⋯ 𝑎 𝑚𝑛 𝑚×𝑛 This matrix is denoted in a simple form as 𝐴 = 𝑎𝑖𝑗 𝑚×𝑛 Where 𝑎𝑖𝑗 is the element in the 𝑖 𝑡ℎ row and 𝑗 𝑡ℎ column
- 4. Type Of Matrices: 1. Row Matrix : A matrix which has only one row is called a row matrix or row vector. i.e. 1 −1 2 3 1×4 2 5 1×2 2. Column Matrix : A matrix which has only one column is called a column matrix or column vector. i.e. 1 2 3 3×1 −1 1 0 4 4×1
- 5. 3. Square Matrix: A matrix in which the number of rows is equal to the number of columns is called a square matrix. i.e. 1 −1 2 3 1 4 5 0 6 3×3 1 2 −1 4 2×2 4. Null or Zero Matrix : A matrix in which each element is equal to zero is called a null matrix and is denoted by O. i.e. 0 0 0 0 0 0 0 0 0 0 0 0 0
- 6. 5. Diagonal Matrix: A square matrix is called a diagonal matrix if all its non diagonal elements are zero. i.e. 1 0 0 0 2 0 0 0 4 3 0 0 5 6. Scalar Matrix : A diagonal matrix whose all diagonal elements are equal is called a scalar matrix. i.e. 3 0 0 0 3 0 0 0 3 5 0 0 5
- 7. 7. Identity orUnit Matrix : A diagonal matrix whose all diagonal elements are unity (1) is called a unit or identity matrix and is denoted by I. i.e. 1 0 0 0 1 0 0 0 1 1 0 0 1 8. Upper Triangular Matrix : A square matrix in which all the entries below the diagonal are zero is called an upper triangular matrix. i.e. 4 −3 2 0 3 5 0 0 1 1 3 0 2
- 8. 10. Trace ofa Square Matrix : The sum of all the diagonal elements of a square matrix is called the trace of a matrix. i.e. If A = −1 2 3 5 7 0 −8 4 1 2 4 −2 7 −3 1 0 then Trace of 𝐴 = −1 + 5 + 7 + 0 = 11 Note : If 𝐴 = 𝑎𝑖𝑗 𝑛×𝑛 then Trace of 𝐴 = 𝑎11 + 𝑎22 + 𝑎33 + ------ +𝑎 𝑚𝑛 = 𝑖=1 𝑛 𝑎𝑖𝑖
- 9. 11. Transpose ofa Matrix : The matrix obtained by interchanging the rows and columns of a given matrix A is called the transpose of A and is denoted by 𝐴 𝑇 or 𝐴′. i.e. 𝐼𝐹 𝐴 = 1 2 3 4 5 6 2×3 𝑡ℎ𝑒𝑛 𝐴 𝑇 = 1 4 2 5 3 6 3×2 Properties : I. 𝐴 𝑇 𝑇 = 𝐴 II. 𝐴𝐵 𝑇 = 𝐵 𝑇 𝐴 𝑇 III. 𝐴 + 𝐵 𝑇 = 𝐴 𝑇 + 𝐵 𝑇 IV. 𝐾𝐴 𝑇 = 𝐾𝐴 𝑇 where K is any scalar.
- 10. 12. Determinant ofa Matrix : If A is a square matrix then determinant of A is represented as 𝐴 or det(A). i.e. 𝐼𝐹 𝐴 = 2 1 −3 4 3 5 5 4 9 𝑡ℎ𝑒𝑛 det 𝐴 = 2 1 −3 4 3 5 5 4 9 13. Singular and Non-singular Matrix : A square matrix A is called singular if det(A)=0 and non-singular or invertible if det(A)≠0. i.e. 𝐿𝑒𝑡 𝐴 = 2 1 −3 4 3 5 5 4 9 𝑡ℎ𝑒𝑛 det 𝐴 = 2 1 −3 4 3 5 5 4 9 ⟹ 𝐴 = 2 27 − 20 − 1 36 − 25 − 3(16 − 15) = 14 − 11 − 3 = 0 Hence, A is a singular matrix.
- 11. Elementary Transformations : Elementary Row Transformations : 1) 𝑅𝑖𝑗 ∶ Interchange of 𝑖 𝑡ℎ and 𝑗 𝑡ℎ row. 2) 𝐾𝑅𝑖 ∶ Multiplication of 𝑖 𝑡ℎ rowby K. 3) 𝑅𝑗 + 𝐾𝑅𝑖 ∶ Addition of K times the 𝑖 𝑡ℎ row to the 𝑗 𝑡ℎ row. Elementary column Transformations : 1) 𝐶𝑖𝑗 ∶ Interchange of 𝑖 𝑡ℎ and 𝑗 𝑡ℎ column. 2) 𝐾𝐶𝑖 ∶ Multiplication of 𝑖 𝑡ℎ column by K. 3) 𝐶𝑗 + 𝐾𝐶𝑖 ∶ Addition of K times the 𝑖 𝑡ℎ column to the 𝑗 𝑡ℎ column.
- 12. Equivalent Matrices: Two matrices A and B of same order are said to be equivalent matrices if one of them is obtained from the other by elementary transformation. Symbolically, we can write A ~ B. Row Echelon form : A given matrix is said to be in row echelon form if it satisfies the following properties: I. The zero rows of the matrix if they exists occur below the non-zero rows of the matrix. II. The first non-zero element in any non-zero row of the matrix must be equal to unity (1). We call this a leading 1.
- 13. III. In anytwo successive rows that do not consists entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row. The following matrices are in row echelon form : 1 1 0 0 1 0 0 0 0 , 1 4 −3 7 0 1 6 2 0 0 1 5 , 0 1 2 6 0 0 0 1 −1 0 0 0 0 0 1
- 14. Reduced RowEchelon Form : A given matric is said to be in reduced row echelon form if it satisfies the following properties : I. A matrix is of necessity in row echelon form. II. Each column that contains a leading 1 has zero everywhere else in that column. The following matrices are in reduced row echelon form. 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 4 0 1 0 7 0 0 1 −1 0 1 0 0 −2 0 1 0 1 3 0 0 0 0 0 0 0 0 0 0
- 15. Notes: (1) Every matrixhas an unique reduced row echelon form. (2) A row-echelon form of a given matrix is not unique. (Different sequences of row operations can produce different row-echelon forms.)
- 16. System of Non-HomogeneousLinear Equations : A system of m linear equations in n unknowns can be written as 𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ ⋯ ⋯ + 𝑎1𝑛 𝑥 𝑛 = 𝑏1 𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ ⋯ ⋯ + 𝑎2𝑛 𝑥 𝑛 = 𝑏2 ⋮ ⋮ ⋮ = ⋮ 𝑎 𝑚1 𝑥1 + 𝑎 𝑚2 𝑥2 + ⋯ ⋯ ⋯ + 𝑎 𝑚𝑛 𝑥 𝑛 = 𝑏 𝑚 The above system can be written in a matrix form as 𝑎11 𝑎12 𝑎21 𝑎22 ⋯ 𝑎1𝑛 ⋯ 𝑎2𝑛 ⋮ ⋮ 𝑎 𝑚1 𝑎 𝑚2 ⋮ ⋯ 𝑎 𝑚𝑛 𝑥1 𝑥2 ⋮ 𝑥 𝑛 = 𝑏1 𝑏2 ⋮ 𝑏 𝑚 Or Simply A X = B
- 17. Where, A = 𝑎11𝑎12 𝑎21 𝑎22 ⋯ 𝑎1𝑛 ⋯ 𝑎2𝑛 ⋮ ⋮ 𝑎 𝑚1 𝑎 𝑚2 ⋮ ⋯ 𝑎 𝑚𝑛 is called coefficient matrix of order m×n. 𝑋 = 𝑥1 𝑥2 ⋮ 𝑥 𝑛 𝑎𝑛𝑑 𝐵 = 𝑏1 𝑏2 ⋮ 𝑏 𝑚 is any vector of order n × 1 & 𝑚 × 1 respectively. 𝐴 : 𝐵 = 𝑎11 𝑎12 𝑎21 𝑎22 ⋯ 𝑎1𝑛 : 𝑏1 ⋯ 𝑎2𝑛 : 𝑏2 ⋮ ⋮ 𝑎 𝑚1 𝑎 𝑚2 ⋮ ∶ ⋮ ⋯ 𝑎 𝑚𝑛 : 𝑏 𝑚 is the augmented matrix of the given system of linear equations.
- 18. Solution of systemof Linear Equations : For a system of m linear equations in n unknowns, there are three possibilities of the solutions to the system I. The system has unique solution. II. The system has infinite solutions. III. The solution has no solution. When the system of linear equations has one or more solutions, the system is said to be consistent otherwise it is inconsistent.
- 19. 1. Gauss-Jordan Elimination: Reducing the augmented matrix to reduced row echelon form is called Gauss-Jordan Elimination. 2. Gauss-Elimination : Reducing the augmented matrix to “row echelon form” and then stopping is called Gaussian Elimination.
- 20. (4) For asquare matrix, the entries a11, a22, …, ann are called the main diagonal entries. 1.2 Gaussian Elimination and Gauss-Jordan Elimination mn matrix: mnmmm n n n aaaa aaaa aaaa aaaa 321 3333231 2232221 1131211 rowsm columnsn (3) If , then the matrix is called square of order n.nm Notes: (1) Every entry aij in a matrix is a number. (2) A matrix with m rows and n columns is said to be of size mn .
- 21. a systemof m equations in n variables: mnmnmmm nn nn nn bxaxaxaxa bxaxaxaxa bxaxaxaxa bxaxaxaxa 332211 33333232131 22323222121 11313212111 mnmmm n n n aaaa aaaa aaaa aaaa A 321 3333231 2232221 1131211 mb b b b 2 1 nx x x x 2 1 bAx Matrix form:
- 22.
- 23. Elementary rowoperation: jiij RRr :(1) Interchange two rows. ii k i RRkr )(:)((2) Multiply a row by a nonzero constant. jji k ij RRRkr )(:)((3) Add a multiple of a row to another row. Row equivalent: Two matrices are said to be row equivalent if one can be obtained from the other by a finite sequence of elementary row operation.
- 24. Ex 2:(Elementary row operation) 1432 4310 3021 1432 3021 4310 12r 2125 0331 1321 2125 0331 2642 )( 1 2 1 r 81330 1230 3421 2512 1230 3421 )2( 13 r
- 25. Ex 3:Using elementary row operations to solve a system 17552 53 932 zyx zy zyx 17552 43 932 zyx yx zyx Linear System 17552 4031 9321 17552 5310 9321 Associated Augemented Matrix Elementary Row Operation 1110 5310 9321 221 )1( 12 )1(: RRRr 331 )2( 13 )2(: RRRr 1 53 932 zy zy zyx
- 26. 332 )1( 23 )1(: RRRr Linear System 4200 5310 9321 2 1 1 z y x 2100 5310 9321 Associated Augemented Matrix Elementary Row Operation 42 53 932 z zy zyx 33 ) 2 1 ( 3 ) 2 1 (: RRr 2 53 932 z zy zyx
- 27. (1) All rowconsisting entirely of zeros occur at the bottom of the matrix. (2) For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). (3) For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row. Reduced row-echelon form: (4) Every column that has a leading 1 has zeros in every position above and below its leading 1.
- 28. form)echelon -row(reduced form) echelon-(row Ex 4:(Row-echelon form or reduced row-echelon form) 10000 41000 23100 31251 0000 3100 5010 0000 3100 2010 1001 3100 1120 4321 4210 0000 2121 form) echelon-(row form)echelon -row(reduced 2100 3010 4121
- 29. Solutions of asystem of linear Equations : There are only two possibilities for the solution of homogenous linear system. I. The system has exactly one solution. i.e. 𝑥1 = 0, 𝑥2 = 0, ⋯ ⋯ ⋯ , 𝑥 𝑛 = 0 This solution is called the trivial solution. II. The system has infinite solutions, this solution is called the non-trivial solution. Note : The system of equation has non-trivial solution if det(A) = 0.
- 30. Cramer’s rule : Theorem-1 : If A X = B is a system of n linear equations in n unknowns such that det(A) ≠ 0, then the system has a unique solution. This solution is 𝑥1 = det(𝐴1) det(𝐴) , 𝑥2 = det(𝐴2) det(𝐴) , ⋯ ⋯ ⋯ , 𝑥 𝑛 = det(𝐴 𝑛) det(𝐴) where 𝐴𝑗 is the matrix obtained by replacing the entries in the 𝑗 𝑡ℎ column of A by the entries in the matrix. B = 𝑏1 𝑏2 ⋮ 𝑏 𝑛 Note : Cramer’s rule can’t be used for a system AX=B in which det(A) = 0.
- 31. Determinant of anUpper Triangular Matrix : Let A = 𝑎11 𝑎12 0 𝑎22 𝑎13 𝑎14 𝑎23 𝑎24 0 0 0 0 𝑎33 𝑎34 0 𝑎44 det (A) = 𝑎11 𝑎12 0 𝑎22 𝑎13 𝑎14 𝑎23 𝑎24 0 0 0 0 𝑎33 𝑎34 0 𝑎44 = 𝑎11 𝑎22 𝑎23 𝑎24 0 𝑎33 𝑎34 0 0 𝑎44 = 𝑎11 𝑎22 𝑎33 𝑎34 0 𝑎44 = 𝑎11 ∗ 𝑎22 ∗ 𝑎33 ∗ 𝑎44
- 32. Theorem-2 : IfA is an 𝑛 × 𝑛 triangular matrix ( upper triangular, lower triangular or diagonal ) then det(A) is the product of the entries on the main diagonal of the matrix. That is det (A) = 𝑎11 ∗ 𝑎22 ∗ 𝑎33 ⋯ ⋯ ⋯ 𝑎 𝑚𝑛 i.e. If A = 2 7 0 −3 −3 8 7 5 0 0 0 0 6 7 0 9 then det(A) = (2) (−3) (6) (9) = −324
- 33. Minors and Cofactors: 1. Minor of an element of a Determinant : If det A = 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 , 𝑡ℎ𝑒𝑛 Minor of 𝑎11 = 𝑎22 𝑎23 𝑎32 𝑎33 Minor of 𝑎22 = 𝑎11 𝑎13 𝑎31 𝑎33 Minor of 𝑎32 = 𝑎11 𝑎13 𝑎21 𝑎23
- 34. 2. Cofactor ofan element of a Determinant : If det A = 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 , 𝑡ℎ𝑒𝑛 Cofactor of 𝑎11 = −1 1+1 𝑎22 𝑎23 𝑎32 𝑎33 Cofactor of 𝑎22 = −1 1+1 𝑎11 𝑎13 𝑎31 𝑎33 Cofactor of 𝑎32 = −1 1+1 𝑎11 𝑎13 𝑎21 𝑎23
- 35. Adjoint of squareMatrix : The transpose of the matrix of the cofactors is called the adjoint of the matrix. 𝐿𝑒𝑡 𝐴 = 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 The matrix formed by the cofactors of the element of A is 𝐶𝐴 = 𝐴11 𝐴12 𝐴13 𝐴21 𝐴22 𝐴23 𝐴31 𝐴32 𝐴33 where 𝐴𝑖𝑗 is the cofactor of 𝑎𝑖𝑗. Then, adj A = 𝐶𝐴 𝑇 = 𝐴11 𝐴21 𝐴31 𝐴12 𝐴22 𝐴32 𝐴13 𝐴23 𝐴33
- 36. Invertible Matrix : IfA is a square matrix and if a matrix B of the same order can be found such that AB = BA = I then A is said to be invertible and B is called an inverse of A. i.e. 𝐵 = 𝐴−1.
- 37. Inverse of aMatrix by Determinant Method : If A is non-singular square matrix, then 𝐴−1 = 1 det(𝐴) 𝑎𝑑𝑗 𝐴 Note : (1) If det (A) = 0 then A is not invertible. i.e. 𝐴−1 does not exist. (2) If A = 𝑎 𝑏 𝑐 𝑑 𝑡ℎ𝑒𝑛 𝑎𝑑𝑗 𝐴 = 𝑑 −𝑏 −𝑐 𝑎 ∴ 𝐴−1 = 1 𝑎𝑑−𝑏𝑐 𝑑 −𝑏 −𝑐 𝑎
- 38. Inverse of amatrix by Elementary Transformation : Gauss-Jordan Method : Let A be a given non-singular matrix of order n. Let 𝐼 𝑛 be the unit matrix of order n. To find 𝐴−1, take the matrix 𝐴 ∶ 𝐼 𝑛 and reduce it with the help of a series of row operations to the form 𝐼 𝑛 ∶ 𝐵 . Then 𝐵 = 𝐴−1.
- 39. Inverses and Powersof Diagonal Matrices : A general n× 𝑛 diagonal matrix D can be written as D = 𝑑1 0 0 𝑑2 ⋯ 0 ⋯ 0 ⋮ ⋮ 0 0 ⋮ ⋯ 𝑑 𝑛 A diagonal matrix is invertible if and only if all of its diagonal entries are non-zero. In this case D is 𝐷−1 = 1 𝑑1 0 0 1 𝑑2 ⋯ 0 ⋯ 0 ⋮ ⋮ 0 0 ⋮ 0 1 𝑑 𝑛
- 40. 𝐷 𝐾 = 𝑑1 𝐾 0 0𝑑2 𝐾 ⋯ 0 ⋯ 0 ⋮ ⋮ 0 0 ⋮ 0 𝑑 𝑛 𝐾 i.e. IF A = 1 0 0 0 −3 0 0 0 2 , then 𝐴−1 = 1 0 0 0 −1 3 0 0 0 1 2 𝐴5 = 1 0 0 0 −243 0 0 0 32
- 41. Rank of Matrix: A matrix is said to be of rank r if it satisfies the following properties : I. There is at least one minor of order r which is not zero II. Every minor of order (r+1) is zero. Rank of matrix A is denoted by 𝜌(𝐴). There are three methods for finding the rank of a matrix 1. Rank of a matrix by Determinant method. 2. Rank of a matrix by row Echelon form. 3. Rank of a matrix by Normal form.
- 42. Note : I.If A is zero matrix then 𝜌 𝐴 = 0. II. If A is not a zero matrix then 𝜌 𝐴 ≥ 1. III. If A is a non-singular n× 𝑛 matrix then 𝜌 𝐴 = 𝑛. IV. 𝜌 𝐼 𝑛 = 𝑛 , where 𝐼 𝑛 is 𝑛 × 𝑛 unit matrix . V. If A is 𝑚 × 𝑛 matrix then 𝜌 𝐴 ≤ 𝑚𝑖𝑛 𝑚 , 𝑛 .
- 43. Rank of Matrixby Row Echelon Form : The rank of a matrix in row echelon form is equal to the number of non-zero rows of the matrix. i.e. Rank of matrix = Number of non-zero rows e.g. 𝐼𝑓 𝐴 = 1 2 0 0 5 0 1 0 0 0 0 0 𝑡ℎ𝑒𝑛 𝜌 𝐴 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑛 − 𝑧𝑒𝑟𝑜 𝑟𝑜𝑤𝑠 𝜌 𝐴 = 2
- 44. Rank of Matrixby Normal Form : If a matrix A is of the form 𝐼𝑟 0 0 0 𝑜𝑟 𝐼𝑟 0 𝑜𝑟 𝐼𝑟 0 𝑜𝑟 𝐼𝑟 Then it called normal form of A. Note : Rank of A = r. SOME SPECIAL MATRIX : 1. Symmetric Matrix : A Square matrix A= 𝑎𝑖𝑗 𝑛×𝑛 is called symmetric matrix if 𝑎𝑖𝑗 = 𝑎𝑗𝑖 ∀ 𝑖 & 𝑗. OR A = 𝐴 𝑇 i.e. 7 −3 −3 5 1 4 5 4 −3 0 5 0 7
- 45. 2. Skew SymmetricMatrix : A Square matrix A= 𝑎𝑖𝑗 𝑛×𝑛 is called Skew symmetric matrix if 𝑎𝑖𝑗 = −𝑎𝑗𝑖 ∀ 𝑖 & 𝑗. OR A = −𝐴 𝑇 If 𝑖 = 𝑗 , 𝑎𝑖𝑖 = −𝑎𝑖𝑖 2𝑎𝑖𝑖 = 0 𝑎𝑖𝑖 = 0 ∀ 𝑖 Thus the diagonal element of a skew symmetric matrix are all zero. i.e. 0 1 4 −1 0 −3 −4 3 0 0 ℎ 𝑔 −ℎ 0 𝑓 −𝑔 −𝑓 0 3. Orthogonal Matrix : A square matrix A is called orthogonal if 𝐴 𝑇 = 𝐴−1 𝑜𝑟 𝐴𝐴 𝑇 = 𝐴 𝑇 𝐴 = 𝐼.
- 46. Complex Matrix : Ifall the element of a matrix are real numbers then it is called a real matrix. If at least one element of a matrix is a complex number 𝑎 + 𝑖𝑏 where 𝑎 , 𝑏 are real numbers and 𝑖 = −1 then the matrix is called a complex matrix. i.e. 2 + 𝑖 0 −5 3 −𝑖 4 + 2𝑖
- 47. Conjugate ofa matrix : The matrix obtained from any given matrix A by replacing its element by the corresponding complex conjugates is called the conjugate of A and is denoted by 𝐴. i.e. 𝐴 = 2 + 3𝑖 −4𝑖 5 7 − 2𝑖 ⇒ 𝐴 = 2 − 3𝑖 4𝑖 5 7 + 2𝑖 Transposed conjugate of a matrix : The transpose of the conjugate matrix of a matrix A is called the transposed conjugate of A and is denoted by 𝐴∗ i.e. 𝐴∗ = 𝐴 𝑇 = 𝐴 𝑇
- 48. Hermitian Matrix : Asquare matrix 𝐴 = 𝑎𝑖𝑗 is said to be Hermitian if 𝑎𝑖𝑗 = 𝑎𝑖𝑗 ∀ 𝑖 & 𝑗 Or 𝐴 = 𝐴∗ where 𝐴∗ = 𝐴 𝑇 i.e. 1 2 − 3𝑖 4 + 𝑖 2 + 3𝑖 3 1 + 2𝑖 4 − 𝑖 1 − 2𝑖 7 Note : In a Hermitian matrix, the diagonal elements are real
- 49. Skew Hermitian Matrix: A square matrix 𝐴 = 𝑎𝑖𝑗 is said to be skew Hermitian matrix if 𝑎𝑖𝑗 = −𝑎𝑖𝑗 ∀ 𝑖 & 𝑗 Or 𝐴 = 𝐴∗ where 𝐴∗ = 𝐴 𝑇 i.e. 2𝑖 3 + 𝑖 5 −3 + 𝑖 0 −1 − 𝑖 −5 1 − 𝑖 −𝑖 Note : In a skew Hermitian matrix , the diagonal elements are zero or purely imaginary numbers.
- 50. Unitary Matrix: A square matrix A is said to be unitary if 𝐴∗ = 𝐴−1 or 𝐴∗ 𝐴 = 𝐴𝐴∗ = 𝐼 Where 𝐴∗ = 𝐴 𝑇
- 51.