Unit Two B.Tech Ist Semester Content Elementary Transformations (Operation) Rank of The matrix Normal Form Solution of System of Linear Eqn. Vectors Linear dependence & independence Linear &Orthogonal transformation Content Eigen value, Eigen vectors and there properties Cayley-Hamilton Theorem Diagonal Form of Matrix Similar Matrix Quadratic form of Matrix Matrix A matrix is a rectangular array of numbers. We subscript entries to tell their location in the array aij " i th row" " j th column" rows m× n row a11 a12 a13 K a1n a a22 a23 K a2 n 21 A = a31 a32 a33 K a3n Matrices are M M M identified by their size. am1 am 2 am 3 K amn Matrix 1× 5 [3 − 1 5 0 2] Row Matrix 4× 1 4× 4 − 2 2 − 1 − 2 4 6 −1 3 Column Matrix 5 7 1 − 2 5 − 8 9 4 7 9 0 − 3 Square Matrix Matrix A matrix that has the same number of rows as columns is called a square matrix. a11 a12 a13 a14 a a22 a23 a24 A= 21 a31 a32 a33 a34 a41 a42 a43 a44 Matrix Diagonal Matrix Unit Matrix a11 0 0 ............0 1 0 0 ............0 0 a 0 ............0 0 1 0 ............0 22 0 0 a33 ...........0 0 0 1............0 ........................... ........................... ........................... ........................... 0 0 0 0 ann n×n 0 0 0 ........ 0 1 n×n 0 0 0 ............0 0 0 0 ............0 0 0 0............0 Null Matrix ........................... ........................... 0 0 0 ........ 0 0 n×n Determinant A determinant is a real number associated with every square matrix. The determinant of a square matrix A is denoted by "det A" or | A |. a b The determinant of 2x2 matrix is = ad − bc c d and 3x3 matrix is a1 a2 a3 b2 b3 b3 b1 b1 b2 b1 b2 b3 = a1 − a2 + a3 c 2 c2 c3 c1 c1 c2 c1 c2 c3 e.g: 1 2 1 − 1 3 4 (a) Find the determinant of the matrix . 5 2 1 a a2 a3 −1 (b) If b b2 b 3 − 1 = 0 in which a,b,c are different, c c2 c3 − 1 show that abc = 1 . Properties of the determinant: i. A determinant of a matrix is unaffected by changing its rows(columns) into columns (rows). ii. If two rows(columns) of a determinant are interchange then the numerical value of determinant remains same but change its sign. iii. If two rows(columns) of a determinant are identical then the value of determinant is zero. iv. If each element of any row(column) are multiplied by the same factor ,then whole determinant multiplied by same factor. v. If each element of a column consist m terms ,the determinant can be expressed as the sum of m determinants. Elementary Transformations Operations that can be performed without altering the solution set of a linear system 1. Interchange any two rows 2. Multiply every element in a row by a nonzero constant 3. Add elements of one row to corresponding elements of another row Some Result of E-operation i. Any E-row operation on the product of two matrices is equivalent to the same operation on the pre-factor. i.e. for E-row operation R, R(AB)=R(A).B . ii. Any E-column operation on the product of two matrices is equivalent to the same operation on the post-factor. i.e. for E-column operation C, C(AB)=A.C(B). Inverse of a Matrix If A is a square matrix s.t. |A|≠0, then a matrix B is called the inverse of the matrix A if AB=BA=I (identity matrix) It is denoted by A-1 . Inverse of a matrix by E-operation (Gauss-Jordan method): Let A be a square matrix s.t. |A|≠0, then A=IA Appling E-row operation on A=IA, to reduce L.H.S. matrix A in to Identity matrix and in R.H.S. Identity matrix I reduced in a new matrix B(say).i.e. R(A)=R(I).A reduced to I=BA…………………(i) Post multiplying (i) by A-1 , we have A-1 =B e.g. : Use Gauss-Jordan method to find the inverse of the 2 0 − 1 matrix 5 1 0 . 0 1 3 Echelon Form All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes, and The leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is 1 and always strictly to the right of the leading coefficient of the row above it. All entries in the column below a leading 1 are zero. 1 # # # 0 1 # # 0 0 1 # e.g.: To change the Matrix A in echelon form by elementary transformation. 5 3 14 4 0 1 2 1 A= 1 − 1 2 0 Minor of Matrix If A be mxn matrix then determinant of sub matrix of a matrix A is called the Minor of the matrix. e.g.: Find all the minors of the matrix 2 1 − 3 − 6 3 − 3 1 2 1 1 1 2 . Rank of the Matrix The order of the highest non-zero minor of a mxn matrix is called Rank of the matrix and is denoted by ρ(A). Note: Number of non-zero row in the echelon form of the matrix is also the rank of matrix. e.g.: (a) Determine the rank of the following matrix: 0 1 −3 − 1 1 0 1 1 3 1 0 1 1 1 −2 0 (b) Reduce the following matrix in Normal Form: 2 1 − 3 − 6 3 − 3 1 2 1 1 1 2 Normal form of a Matrix If A is an mxn matrix and by a series of E-operation, it can be find in the following form I r 0 I r 0 , 0 , [I r 0], [I r ] ,where Ir is the unit 0 matrix of order r. Note: For mxn matrix A of rank r, to find square matrices P & Q of order m and n respectively, such that PAQ is in normal form. P and Q can be formed by applying E-operation on A=IAI. 1 1 2 e.g.: For the matrix A = 1 2 3 find the non- 0 − 1 − 1 singular matrix P and Q such that PAQ is in normal form. Solution of a system of linear Equation a1 x + b1 y + c1 x = d1 Consider of a system of linear eqn . a2 x + b2 y + c2 x = d 2 In matrix notation we have a3 x + b3 y + c3 x = d 3 AX = B …………..(i) a1 b1 c1 Where A = a2 b2 c2 is called the co-efficient matrix. a3 b3 c3 d1 x d is determine X = y is variable matrix and B = 2 matrix. z d 2 If B≠0 then eqn. (i) is called non-homogeneous system of equation, if B=0 then (i) is homogeneous system of equation. Solution of non-homogeneous system of linear eqn. For a system of non-homogeneous linear equation AX=B i. If ρ[A:B]≠ρ(A), the eqn. have no solution, system is called inconsistent. ii. If ρ[A:B]=ρ(A)= number of variable, the system has a unique solution. iii. If ρ[A:B]=ρ(A)< number of variables, the system has an infinite no. of solutions. Solution of homogeneous system of linear eqn. For a system of homogeneous linear equations AX=0 X=0 is always a solution, is called trivial solution. Thus a homogeneous system is always consistent. A system of homogeneous linear equation has either the trivial solution or infinite number of solutions. (i) if ρ(A)= number of variable, the system has only trivial solution. (ii) if ρ(A)< number of variable, the system has an infinite number of non-trivial solution. e.g: a) Test the consistency of the equation x+2y-z=3,3x- y+2z=1, 2x-3y+7z=5, 2x-2y+3z=2, x-y+z=-1, and solve them. b) For what values of λ the equation 3x-y+4z=3, x+2y- 3z=-2, 6x+5y+ λz=-3 have (i) no solution. (ii) a unique solution (iii) an infinite number of solution. Vector Any ordered n-tuple of numbers is called an n-vector. If x1 , x2 , ................xn be any n numbers then the ordred n-tuple X = ( x1 , x2 ,...............xn ) in called n-vector. e.g.: In 3D the coordinates are represent with 3- vectors (x,y,z). Linearly dependent and independent of vector.: A set of r vectors X1, X2, …………., Xr is said to be linearly dependent if there exist r scalars k1, k2, ………., kr not all zero , such that k1 X 1 + k 2 X 2 + ......... + k r X r = 0 and X1, X2, …………., Xr is said to be linearly independent, ∀ k1 X 1 + k 2 X 2 + ......... + k r X r = 0 ⇒ k1 = k 2 = ......... = k r = 0 Note: Let X1, X2, …………., Xr are linearly dependent vectors, such that, for r scalars k1, k2, ………., kr not all zero k1 X 1 + k 2 X 2 + ......... + k r X r = 0 if ki ≠ 0 , then ki X i = − k1 X 1 − k 2 X 2 − ..... − ki −1 X i −1 − ki +1 X i +1....... − k r X r k1 k2 ki −1 ki +1 kr ⇒ X i = − X 1 − X 2 − ........ − X i −1 − X i +1...... X r ki ki ki ki ki then Xi is a linear combination of the remaining vectors k1, k2, ……… ki-1, ki+1 ,………..kr . Linear Transformation If T : R n → R m is a linear transformation, then there exist a unique mxn matrix A, s.t. T(X)=AX for all X ∈ Rn . Let P(x,y) in a plane transformation, transform to the point P’(x’,y’) under the reflection in co-ordinate axes, then the co-ordinate of P’ can be expressed in terms of those of P by the linear transformation of the form x' = a1 x + b1 y or in matrix notation x' = a1 b1 x y ' a b y y ' = a2 x + b2 y or X’=AX . 2 2 Orthogonal Transform The linear transformation Y=AX, where y1 a11 a12 .......... a1n x1 y a x : , A = 21 a22 .......... 2 a2 n , X = :2 Y= : ...... ....... ........... ..... : yn am1 am 2 ........... amn xn is said to be orthogonal if it transformation 2 2 2 2 2 2 y1 + y2 + ........... + yn in to x1 + x2 + .............. + xn . Note : If the transformation matrix is non-singular , i.e. if |A|≠0, then the linear transformation is called non- singular or regular transformation, if A is singular, i.e. |A|=0 then is called singular. e.g.: Show that the transformations y1 = x1 − x2 + x3 y2 = 3 x1 − x2 + 2 x3 y3 = 2 x1 − 2 x2 + 3 x3 is non-singular. Find the inverse transformation. Eigen value Consider a linear transformation Y=AX. If Y be a matrix of some scalar multiple(say λ) of X, i.e. Y= λX , then AX = λX ⇒ AX − λIX = 0 ⇒ ( A − λI )X = 0...................................(i ) eqn. (i) is system of homogeneous linear eqn. These system will have non-trivial solution if the coefficient matrix is singular, i.e. | A − λI |= 0...............(ii ) eqn. (ii) are called Characteristic Equation. and having n roots ,say λ1, λ2,……….., λn which are called Eigen value or Characteristic Roots or Latent Roots. Eigen vectors For every value of λ, the homogeneous system of eqn.(i) have non-zero solution a1 a 2 . X = which is called an Eigen vector corresponding . . to that value of λ. an e.g.: Find the Eigen value and Eigen vector of the matrix. − 2 2 − 3 2 1 − 6 (i) −1 − 2 0 8 −6 2 (ii) − 6 7 − 4 2 −1 3 Properties of Eigen Value. (i) The eigen value of square matrix A and its transpose A’ are the same. (ii) The sum of the eigen values of a matrix is the sum of the elements on principal diagonals. (iii) The product of the eigen values of a matrix A is equal to |A|. 1 (iv) If λ is an eigen value of a non-singular matrix A, then is an eigen value of of A-1 . λ (v) If λ is an eigen value of an orthogonal matrix A, then is 1 an eigen value of of A-1 . λ (vi) If λ1 , λ2 ,........, λn are the eigenmvalues of a matrix A, then A m has the eigen values λ m , λ ,........, λ. m 1 2 n where m∈ Z + Cayley Hamilton Theorem Every square matrix satisfies it’s characteristic equation. i.e. if the characteristic equation of the nth order matrix A is , | A − λI |= (−1) n λn + a1λn −1 + a2 λn − 2 + ............. + an = 0 then (−1) n An + a1 An −1 + a2 An −2 + ............. + an I = 0 Note: According to Cayley Hamilton theorem (−1) n An + a1 An −1 + a2 An − 2 + ............. + an I = 0 Multiplying this by A-1 we have (−1) n An −1 + a1 An − 2 + a2 An −3 + ............. + an −1 I + an A−1 = 0 1 ⇒ A = − [(−1) n An −1 + a1 An − 2 + a2 An −3 + ............. + an −1 I ] −1 an e.g.: (a)Verify Cayley Hamilton theorem for the matrix A and find A-1 7 2 − 2 where A = − 6 − 1 2 . 6 2 − 1 1 4 (b) Find the characteristic equation of the matrix A = 2 3 and use it to find the matrix represented by A5 + 5 A4 − 6 A3 + 2 A2 − 4 A + 7 I Also express A5 − 4 A4 − 7 A3 + 11A2 − A − 10 I as a linear polynomial in A. Diagonalization of Matrix If a square matrix A of order n has n linearly independent eigen vectors then there exist a matrix B such that B- 1AB is a diagonal matrix. 8 −6 2 A = − 6 7 − 4 e.g.: Diagonalise the matrix 2 −4 3 Similar Matrix Let A and B be square matrices of the same order. The matrix A is said to be similar to the matrix B if there exist an invertible square matrix P, s.t. A=P-1BP. e.g.: Show that the matrices 5 5 1 2 A= and B = are similar. − 2 0 − 3 4 Quadratic Form A real quadratic form is an homogeneous expression of the form n n Q = ∑∑ aij xi x j in which the total i =1 j =1 power in each term is 2. For instance, if a h f x A = h b g , X = y and X ' = [x y z ] f g c z then X ' AX = ax 2 + by 2 + cz 2 + 2hxy + 2 gxz + 2 fyz which is quadratic form. Canonical Form n n If the quadratic form X ' AX = ∑∑ aij xi x j can be reduced to i =1 j =1 n the quadratic form Y ' BY = ∑ λi yi 2 by a non-singular linear i =1 transformation X=PY then Y’BY is called the canonical form of given one. If B = P' AP = diag (λ1 , λ2 ,............, λn ) , then n X ' AX = Y ' BY = ∑ λi yi 2 i =1 . Where B=P’AP. Index and Signature of the Quadratic form The number of positive terms in the canonical form is called the index of the quadratic form. Let p be the number of positive terms and rank of matrix is r, then the signature of matrix =2p-r. e.g.: Reduce3x 2 + 5 y 2 + 3z 2 − 2 xy − 2 yz + 2 zx quadratic forms to canonical forms by orthogonal transformation. Write also rank, index and signature. Nature of Quadratic form A real quadratic form X’AX in n variables is said to be i. positive definite if all the eigen values of A is >0. ii. negative definite if all the eigen values of A is <0. iii. positive semi-definite if all the eigen values of A is ≥0 and at least one eigen value =0. iv. negative semi-definite if all the eigen values of A is ≤0 and at least one eigen value =0.
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