TABLE OF CONTENTS
Preliminaries: Determinants.Applications: calculation of the rank of a matrix, calculation of the inverse matrix and Cramer’s Rule
Chapter 1. Diagonalization of square matrices
1.1. Rn: Spanning sets. Basis.
1.2. Eigenvalues and eigenvectors of a square matrix: definition and calculation.
1.3. Diagonalization of a square matrix.
1.4. Application to the calculation of matrix powers.
Chapter 2. Real quadratic forms
2.1. Quadratic forms: definition. Matrix expression and polynomial expression.
2.2. Diagonal expression of a quadratic form.
2.3. Classification of a quadratic form according to its sign.
2.4. Constrained quadratic forms.
Chapter 3. Functions from Rn to Rm
3.1. Preliminaries: topological concepts.
3.2. Functions: domain, range and graph. Level sets of scalar functions.
3.3. Continuity of a function.
3.4. Differentiation of a function. Partial derivatives. Gradient vector. Jacobian matrix.
3.5. Differentiability. Directional derivative of differentiable functions.
3.6. Differentiation of composed functions: Chain’s Rule. Tree diagrams.
3.7. Higher order derivatives. Schwarz’s Theorem. Hessian matrix. Taylor’s Theorem.
3.8. Implicit function Theorem. Differentiation of implicit functions.
3.9. Homogeneous functions. Euler’s Theorem.
3.10. Basic integration methods of function of one variable. Barrow’s Rule.