Hours:
20 hours (5 credits)

Room:

Aula Riunioni del Dipartimento di Ingegneria dell’Informazione, Via G. Caruso 16, Pisa - Ground Floor

To register to the course, click here

Short Abstract:

Several problems in advanced engineering rely on the manipulation of vectors or matrices that are functions of other vectors or matrices, and their derivatives and differentials. Examples arise in various fields spanning Analytical Mechanics, Robot Control, Deep Neural Networks, Optimization and Computer Vision.
Although the such objects are studied since more than a century, modern theory of matrix algebra and matrix calculus introduced, based the Kronecker product and matrix vectorization, simple yet unambiguous, therefore powerful and flexible, definitions of matrix derivatives and matrix differentials. Those tools have been successfully used to obtain novel theoretical results in the fields of Probability, Psychometrics and Econometrics, by enabling the formal manipulation of complex expressions.
This course will introduce such mathematical instruments in detail and present their application to several relevant problems encountered in the aforementioned engineering fields, to streamline their description and yield more elegant and compact derivations, that will hopefully enable advanced student of those fields to achieve novel and significant results.

Course Contents in brief:

1. Matrix vectorization, Kronecker product, Trace and their properties
2. Definition of Matirx Derivatives and Matrix Differentials and some Notable Results
3. Applications to Analytical Mechanics: Rotation differentials, Robots Jacobian and Stiffness
4. Applications to Robot Control: Lagrange Dynamics, Regressor Form of Robot Dynamics
5. Applications to AI: description of a Deep Neural Network, Computer Vision.

Schedule:

June 23, 2025, 9:00-13.00:

1. Matrix vectorization, Kronecker product, Trace and their properties (2h)
2. Magnus definition of derivatives and their application to matrix calculus (1h)
3. Series expansion of a vector field (1h)

June 24, 2025, 9:00-13:00:

1. Derivative of matrix inverse, matrix determinant and matrix pseudo-inverse (2h)
2. Integration of angular speeds into rotation matrices, definition of angular error (1h)
3. Jacobians: algebraic, geometric, and geometric from algebraic (1h)

June 25, 2025, 9:00-13:00:

1. Jacobians: algebraic, geometric, and geometric from algebraic (1h)
2. Stiffness of a kinematic chain (2h)
3. Second Order and Higher Order Differential Kinematics (1h)

June 26, 2025, 9:00-13:00:

1. Lagrangian dynamics with kron and vec (2h)
2. Robot Kinematics and Dynamics in Regressor Form: either and both (2h)

June 27, 2025, 9:00-13:00:

1. Deep Neural Networks and Back-propagation (2h)
2. Derivative of ReLU, MAX, softMAX, and softReLU (1h)
3. Exercise: Neuromorphic Adaptive control with arbitrary gain matrix (1h)