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Simple And Esy Notes On Maths And IC Engine

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Published in: Mathematics
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INFINITE SERIES,LINEAR ALGEBRA,Linear Differential Equations,ORDINARY DIFFERENTIAL EQUATIONS OF FIRST ORDER,COMPLEX NUMBERS AND ELEMENTARY FUNCTIONS OF COMPLEX VARIABLE,LAPLACE TRANSFORM,IC ENGINE MECHANISM,MULTI POINT FUEL INJECTION,PISTON INFORMATION,Testing of IC Engines

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  1. LINEAR ALGEBRA Definition of matrices:- (1) A rectangular array A of mn complex numbers arranged in m rows and n columns is called a matrix of size m xn. A m x n matrix is usually represented in the form . OR more compactly by The entries Clij in the matrix is called element of the matrix. With i and j given, the element in the ith row and the jth column is said to have row index i and column index j. (2) A n x I matrix a (i.e. a matrix having one column only) is called a column vector. An I x n matrix b (i.e. a matrix having one row only) is called a row vector. The norm of a column vector a and a row vector b is defined as: or and they are said to be normalized if (3) (4) and Two m x n matrices elements are equal, i.e. Clij = bij, for all sizes cannot be equal. or are equal if and only if their corresponding Matrices of different A n x n matrix is called a square matrix of order n.
  2. (5) and A matrix, whether square or not, in which every element is zero is called a zero matrix o o o o o o o is denoted by the symbol O, i.e. 2. The algebra of matrices (1) The sum (or difference) of two m n matrices matrix where co = ao + bij (or Cij = aij - bij), for all Note : We usually write C = A + B (or C = A B) (2) The scalar multiplication of am x n matrix number) is a m x n matrix where clij = kaij, for all a a. and is the m n by a scalar k (k is a real or complex and is written as D = kA. The difference between two matrices A and B can be written as A -B = A + (-1)B . For all positive integers m and n, for all scalars k and l, and for all m x n matrices ] and C , we have the following algebraic laws . (iii) A + O = O + A (v) (vii) (k = A) (3) (vi) (Viii) 1 = x p matrix and ap The product of a m , for all x n matrix 4] is a m x n matrix
  3. And is written as C = AB. In general AB BA For all scalar c, and for all m x p matrix A, p x n matrices B and C, m x n matrix D, we have the following algebraic laws (i) c(AB) = (CA)B = A(cB) (ii) + C) = AB + (iii) (B + = BD + CD (iv) A(BC) = (AB)C (4) u 1 u u and For any two column vectors and v is given by the products, 1 of order n, the dot product between u Two vectors u and v are said to be orthogonal if and only if u • v = I] (5) In any square matrix A, the sum of the diagonal elements is called trace and it is denoted by a 11 a m a a a m a 5-1 a a a , then TrA, i.e. if 11 The trace of matrices has the following properties: (i) Tr(AB) = Tr(BA) (ii) Tran) = (TrA)n 3. Special matrices (1) The transpose of am x n matrix where bij = aji. is the n x m matrix For all matrices A, B, and any scalar c, the transposition of matrices has the following properties :- (ii) (CA) T = CAT
  4. (iii) If A ± B is defined (iv) If AB is defined (AB)T = BTAT (2) A square matrix of order n is called an identity matrix of order n if and only if 0 if i J and is denoted by In. For instance: It can be shown that for any square matrix A and identity matrix I , both are of the same order, Al = IA = A (3) A square matrix i.e. Note : Obviously (4) A square matrix (a) symmetric if and only if A of order n is said to be diagonal if dij = 0 for for any positive integer N. of order n is said to be (b) skew-symmetric if and only if AT = - A. (5) The conjugate of a matrix A is the matrix whose elements are, respectively, the conjugates of the elements of A. The matrix defined by
  5. is called the hermitian conjugate of A. (i) The matrix A is said to be hermitian if and only if A (6) (ii) A square matrix A is said to be unitary if (7) Let be a square matrix of order n. The cofactor matrix of A, denoted by cof A, is defined by cof A = whereAij is the cofactor of Clij, for every (8) Let be a square matrix of order n. The transpose of the cofactor of A, i.e. (cof is called the adjoint of A, denoted by adj A. (9) A square matrix A of order n is said to be non-singular or invertible if and only if there exists a square matrix B such that where I is an identity matrix of order n, and the matrix B is called the multiplicative inverse or simply inverse of A, which is denoted byA-l, i.e. AA-I I Note: The inverse of a non-singular matrix is unique. ( 10)Theorem For any square matrix A of order n, A(adj A) = (adj = (det , where I is an identity matrix of order n. det (11) A square matrix A is non-singular if and only if (12) Let A be a square matrix, if det A then A is non-singular and 1 det A (13)) A square matrix A is said to be singular or not invertible if and only if the inverse A-I of A does not exist. Note: A square matrix A is singular if and only if detA = 0. (14) Let A, B are non-singular square matrix of order n and any scalar c, we have
  6. (a) A-I is non-singular and (A ) (b) AB is non-singular and (AB)- -1 (c) An is non-singular for any positive integer n and (A ) -1 (d) A is non-singular for any non-zero scalar c and (e) AT is non-singular and (AT) I = (A Elementary Operations: The three types of elementary operations are these: (El) Interchange Of two rows or columns 1 (E2) Multiplication of (each element of) a row or column by a non zero element k. (E3) Addition of k times the element of a row or column to the corresponding elements of another row or column, Elementary matrices: The matrix obtained from a unit matrix by subjecting it to one of the E-Operations is called an Elementary matrix,. The Rank of a Matrix: It is the order of the largest determinant that can be formed from the elements of the matrix. A matrix A is said to have rank r if it contains at least one square submatrix of r rows with a non- zero determinant, while all square submatrices of (r + 1) rows, or more, have zero determinants. For an r x c matrix, If r is less than c, then the maximum rank of the matrix is r. If r is greater than c, then the maximum rank of the matrix is c. The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one. How to Find Matrix Rank In this section, we describe a method for finding the rank of any matrix. This method assumes familiarity with echelon matrices.
  7. The maximum number of linearly independent vectors in a matrix is equal to the number of non- zero rows in its row echelon matrix . Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows. 1 2 1 2 3 1 1 1 2 Example : Determine the rank of Solution: To determine the row-rank of we proceed as follows. 1 2 1 1. 1 2. 1 o o 3. 1 o o 4. 2 3 1 2 2 1 o o 1 o 1 1 2 1 1 2 1 -1 1 R3(1/2), 2) -1 1 1 1 o 1 o 1 o o 1 o o 2 -1 -1 2 1 o 1 o o 1 o 1 -1 1 1 1 2 1 1 o o 1 The last matrix A which has 3 non-zero rows. Thus, rank(A) =3 Gauss jordan method of finding inverse of matrix : Let A be nxn matrix. The following steps will produce the inverse of A, written A-I. Note the similarity between this method and GAUSS JORDAN method, used to solve a system of equation.
  8. Working Rule: In practice,to find the inverse of A by E-Operations, we write A and I Side by side and the same operations are performed on both.As soon as A is reduced to I, I will reduce to Example : Find the inverse of the matrix 2 1 1 Solution: Consider the matrix 1 2 1 1 1 2 2 1 1 1 o o 1 2 1 o 1 o 1 1 2 using the Gauss-Jordan method. o o 1 A sequence of steps in the Gauss-Jordan method are: 2 1 1 1 1 o o 1 2 1 2 1 o o 1 1 2 1 2 1 1 o o o o o 1 o o 1 o o O RI(1/2) 1 1 o 1 o 1 o 1 R21(-1) R2(2/3) 2 1 o o o o O 1 2 1 o o o o 1 o 1 o 1 o o 1 o o 1 o o 1 o 1 o o o Re ( 1/2) O R3(3/4) O 1 R23 o 1/3) o 1/2) o
  9. o o 1 o o 1 1 R12(-1/2) O o 3/4 o 1 o o 1 3/4 —1/4 3/4 8. Thus, the inverse of the given matrix is System of Linear Algebraic Equations 1 Linear Algebraic Equation : A polynomial equation ' is called a linear algebraic equation. If b the of first degree in the variables equation is said to be nonhomogeneous and b is called the nonhomogeneous term. If b = 0 the equation is said to be homogeneous. Now suppose we have m simultaneous linear equations amX1 = b, The above equations are called a system of linear algebraic equations or a linear system. The numbers ' are called the constants of the system. If these constants are all aeros, the system is said to be homogeneous. A linear system is non homogeneous if and only if at least one of its equations has a non homogeneous term. The elements of the matrix
  10. are called the coefficients of and itself is called coefficient matrix of the system. If we define x and b as the column vectors 1 Ax = h System can be written in the compact matrix form Solution Set : A solution of a linear system is an assignment of values to the variables Xl, x2 xn such that each of the equations is satisfied. The set of all possible solutions is called the solution set A linear system may behave in any one of three possible ways: 1. The system has infinitely many solutions. 2. The system has a single unique solution. 3. The system has no solution. Working rule to check the consistency of non homogeneous system of linear equations: Step(l): find the rank of (A:B) using elementary row operations. if rank(A:B) =rank(A) —No of unknowns ,then system is consistent and has a Step(2): (a)- unique solution. (b)- if rank(A:B)= rank(A) < No of unknowns ,then system is consistent and has infinite number Of solutions. if rank(A) ,then system is inconsistent Working rule to check the consistency of homogeneous system of linear equations: Step(l): find rank of matrix A Step(2) :if rank(A)=No of unknowns, then system is consistent and has trivial solution. Step(3) : rank(A)
  11. Linear dependence and Linear independence of vectors : A Set of r, n-tuple vector Xl,X2,... ... . x r is said to be linearly dependent if there exists r scalars k k .. ..kr not all zero such that klX1+k2X2+ A Set of r, n-tuple vector Xl,X2,... ... relation of the type klX1+k2X2+. krxr -O . x r is said to be linearly independent if every krXr implies kl=k2 kr=0 Orthogonal Matrix : In linear algebra an orthogonal matrix, is a square matrix with real entries whose columns and rows are orthogonal unit vectors Equivalently, a matrix Q is orthogonal if its transpose is equal to its inverse —1 From which QTQ - QQT = 1, where I is the identity matrix. An orthogonal matrix Q is necessarily square and invertible, with inverse Q I = QT. As a linear transformation. Note : If an orthogonal matrix is real, then it is also a normal matrix. Complex Matrices The Conjugate of a Matrix: If the elements of a matrix A are complex quantities, then the matrix obtained from A, on replacing its elements by the corresponding conjugate complex numbers, is said to be the conjugate matrix of A and is denoted by A or A* Properties: • If a is a complex number and A a matrix of any order say m x n, then (aA)*= a *A* 1.3.2 The Conjugate transpose or Hermitian Conjugate of a Matrix: The matrix, which is the conjugate of the transpose of a matrix A is said to be the conjugate 6 transpose of A and denoted by A
  12. Properties: • If a is a complex number and A a matrix, then (aA)6 (AB)6 * A6B6 Hermitian, Skew-Hermitian and Unitary Matrices : A matrix A is said to be (i) Hermitian if A A (ii) skew-Hermitian if 146 — (iii) unitary if A 6 A =A • Characteristic-value problems For a square matrix A in order n all am a a am a a Ix a a which satisfies the following matrix equation Ax = Ax where 1 is a parameter (1) where x is the column vector of order n (2) The above matrix equation is known as the characteristic-value problem (or eigenvalue problem) of the matrix A. We can use the n x n identity matrix I to write Eq.(l) as Ax =11x, or, equivalently, as (a) (A - Il)x = O or (b) (11 - = O Eq.(3)(b) can be satisfied if and only if (3)
  13. X —all a -a a det.(XI — A) = -a When expanded, the determinant (4) p(1) = det(ll - A) (5) becomes a polynomial of degree n in the parameter 1 . Eq.(4) is known as the characteristic equation of the matrix A, and p(1) is called the characteristic polynomial of A. The n roots of p(1) = 0 are called the characteristic values (or eigenvalues) of the matrix A, and the corresponding nontrivial solution vectors of Eq.(3) are called the characteristic vectors. Example Given a square matrix A of order 2 4-5 1-2 (a) Find the characteristic values and the corresponding normalized characteristic vectors of the matrix A. (b) Construct a matrix U of order 2 such that where D is a diagonal matrix of order 2. 4 (c) Using the above results, evaluate A . Solution (a) The characteristic equation is given by 5 x' -2N-3=o
  14. When 1 = 3 Therefore, the normalized characteristic vector for 1 = 3 IS When 1 = -1, Therefore, the normalized characteristic vector for 1 = -1 is (b) Using the results in (a), we get and (LIT-IAU and
  15. Theorem : The characteristic values of a Hermitian matrix A are real. Proof If the characteristic-value problem of matrix A has n characteristic values x.l,x with corresponding non-trivial characteristic vectors For any characteristic vector Xi, we have (1) Axi = IiXi Multiplying both sides of (1) by , we obtain i Taking the Hermitian conjugate of both sides of (2a), we get Ax.i = (2b) Hence, from the results of (2a) and (2b), we have Therefore, all the characteristic values of A are real. Theorem: If the characteristic-value problem of a Hermitian matrix A of order n has n distinct characteristic values vectors with corresponding non-trivial characteristic
  16. Xl,x x. where The characteristic vectors are orthogonal to each other. Proof It is obvious that the characteristic vectors are distinct for distinct characteristic values. For any two distinct characteristic vectors Xi and xj ( ), we have (la) (1b) Axi = IiXi AX] = Ijxj Multiplying both sides of (la) by and both sides of (1b) by , we obtain . Ax.i Ax = X i,.g.x.i (2b) Taking the Hermitian conjugate of both sides of (2a), we get A X. = XiEi X Ax Hence, from the results of (2b) and (2c), we have Cayley Hamilton Theorem : It states that Every square matrix satisfies its own characterstic equation. 1 Example: Verify Cayley-Hamilton theorem for the matrix 2 Il)
  17. Solution : Consider the matrix 21 Its charecteristic equation is given by 1-1 1 p(1) = det (A -Il) = 2 1-1 -2 A-I 32 22 42 10 01 oo oo Hence Every matrix satisfies its own characteristic equation. Diagonalization of Square Matrices One of the reason to consider a characteristic-value problem of a square matrix A of order n is to reduce it into a diagonal matrix Such process is known as the diagonalization of matrix A. If the characteristic-value problem of matrix A has n distinct characteristic values, we have the following theorems
  18. Definition : Two square matrices A and B of order n are said to be unitary equivalent if there exists an unitary matrix U such that B = U*AU. Theorem For any two unitary equivalent matrices A and B have the same characteristic equations and hence they have the same set of characteristic values. Proof By hypothesis, since A is unitary equivalent to B, there exists an unitary matrix U with the property that B = U*AU. The characteristic equation of B is - B) = - U*AU = -A)U = det U •U det.(XI — A) = det.l det.(XJ — A) = det(XJ — A) Theorem If the characteristic-value problem of a Hermitian matrix A of order n has n distinct real characteristic values characteristic vectors 1 (a) U is unitary. with corresponding non-trivial normalized x x x where x We can define a matrix with the following properties: (b) U*AU = D where D is a diagonal matrix
  19. i.e. U can be used to diagonalize the matrix A. Proof (a) To prove that U is unitary, we have to consider [AXI Xl,.r.l ' x x Hence U is unitary. (b) Ah X,x.l
  20. Note: If U is a real matrix, we have U Similar Matrices : Let A and B be square matrices of the same order. The matrix A is said to be similar to B if there exists an invertible matrix P such that A=P-IBP or PA=BP Post multiply both sides by P -l , We have PAP-I = Bl -B So A is similar to B if and only if B is similar to A. The matrix P is called the similarity matrix. Quadratic forms :A homogeneous polynomial of second degree in any number of variables is called a quadratic form.for example: (1) ax2 +2hxy +by 2 (2) ax2 +by2 +cz2 +2hxy +2gyz +2fzx (3) ax2+by2 +c z2+dw2 +2fzx+21xw+2myw+2nzw are quadratic forms in two, three and four variables. Canonical form :lf a real quadratic form be expressed as a sum or difference of the squares of new variables by means of any real non-singular linear transformation, then the later quadratic expression is called a canonical form of given quadratic form. Index and signature of quadratic form : The number p of positive terms in the canonical form is called the index of the quadratic form, and Signature of quadratic form = (The number of +ve terms) (The number of -ve terms)

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