Matrix algebra

Mathematics for Materials and Design

The full set of notes for this topic is here. All materials on this topic can be downloaded here.

Sessions:

1. Introduction to matrices
   • Lecture materials: slides and handout.
   • Videos:
     • Introduction, order and addition.
     • Example - matrix addition and subtraction.
     • Multiplication.
     • Example - scalar multiplication.
     • Example - matrix multiplication.
     • Transpose.
     • MATLAB.


2. Determinants and inverse matrices
   • Lecture materials: slides and handout.
   • Videos:
     • Introduction and determinant.
     • Example - determinant.
     • Inverse matrices.
     • Example - inverse matrices.
     • MATLAB.

     Note that when discussing the role of the determinant in image transformations, we are talking about the absolute value of the determinant. So if for matrix A, -1<det(A)<1 then its effect is area-contracting while if 1<|det(A)| then it is area-expanding.



3. Solving systems of simultaneous linear equations using matrices
   • Lecture materials: slides and handout.
   • Videos:
     • Introduction and theory.
     • Solving systems of simultaneous linear equations - example I.
     • Solving systems of simultaneous linear equations - example II.
     • MATLAB.


4. Vectors and matrices
   • Lecture materials: slides and handout.
   • Videos:
     • Introduction and vectors.
     • What are eigenvalues and eigenvectors?
     • Application to stability analysis.
     • Example - identifying eigenvectors by experimentation.


5. Calculating eigenvalues and eigenvectors
   • Lecture materials: slides and handout.
   • Videos:
     • Introduction and theory.
     • Example - calculating the eigenvalues.
     • Example - calculating the eigenvectors.
     • MATLAB.


6. Eigenvalues and linear stability analysis
   • Lecture materials: slides and handout.
   • Videos:
     • Introduction and theory.
     • Example - calculating eigenvalues and eigenvectors of a 3x3 matrix.
     • Return to the stability analysis application and an example.
     • Example - calculating the eigenvalues in greater detail.