MATRICES | Problem 3| Diagonalization by Orthogonal Transformation | Engineering Mathematics

diagonalization m1, diagonalization of matrices, diagonalization of matrix, diagonalization determinant, diagonalization matrices, diagonalization matrix orthogonal transformation determinant, orthogonal transformation formula, orthogonal transformation in matrix, orthogonal and unitary transformations UNIT – I MATRICES Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties of Eigenvalues and Eigenvectors – Cayley – Hamilton theorem – Diagonalization of matrices by orthogonal transformation – Reduction of a quadratic form to canonical form by orthogonal transformation – Nature of quadratic forms – Applications: Stretching of an elastic membrane. UNIT – II DIFFERENTIAL CALCULUS MA3151 Syllabus Matrix and Calculus Representation of functions – Limit of a function – Continuity – Derivatives – Differentiation rules (sum, product, quotient, chain rules) – Implicit differentiation – Logarithmic differentiation – Applications : Maxima and Minima of functions of one variable. UNIT – III FUNCTIONS OF SEVERAL VARIABLES MA3151 Syllabus Matrix and Calculus Partial differentiation – Homogeneous functions and Euler’s theorem – Total derivative – Change of variables – Jacobians – Partial differentiation of implicit functions – Taylor’s series for functions of two variables – Applications : Maxima and minima of functions of two variables and Lagrange’s method of undetermined multipliers. UNIT – IV INTEGRAL CALCULUS MA3151 Syllabus Matrix and Calculus Definite and Indefinite integrals – Substitution rule – Techniques of Integration: Integration by parts, Trigonometric integrals, Trigonometric substitutions, Integration of rational functions by partial fraction, Integration of irrational functions – Improper integrals – Applications : Hydrostatic force and pressure, moments and centres of mass. UNIT – V MULTIPLE INTEGRALS Double integrals – Change of order of integration – Double integrals in polar coordinates – Area enclosed by plane curves – Triple integrals – Volume of solids – Change of variables in double and triple integrals – Applications : Moments and centres of mass, moment of inertia.