Engineering Mathematics I

Algebra of Matrices

• Addition, scalar multiplication and matrix multiplication.
• Properties include associativity, distributivity and non-commutativity.
• Transpose and conjugate transpose operations.

Diagram: matrix-algebra.png

Rank and Inverse of a Matrix

• Rank is the maximum number of linearly independent rows or columns.
• Determined using row echelon form or determinant methods.
• Inverse exists only if det(A) ≠ 0.

Diagram: rank-echelon.png

Hermitian, Skew Hermitian and Unitary Matrices

• Hermitian: A = A†.
• Skew Hermitian: A = −A†.
• Unitary: A†A = I.
• Important for stability and orthogonality in linear systems.

Eigenvalues and Eigenvectors

• Defined by A x = λ x.
• Obtained from characteristic equation det(A − λI) = 0.
• Used in system stability, diagonalization and modal analysis.

Systems of Linear Equations and Consistency

• Ax = b form.
• Consistent if rank(A) = rank(A b).
• Unique solution if rank(A) = rank(A b) = n.
• Infinite solutions if rank(A) = rank(A b) < n.

Numerical Methods for Solving Systems

• Gauss elimination and Gauss Jordan elimination.
• Gauss Seidel iterative method.

Diagram: gauss-methods.png

Vector Spaces and Linear Transformations

• Vector space: set with vector addition and scalar multiplication.
• Basis: minimal set of linearly independent vectors that span the space.
• Linear transformations preserve vector addition and scalar multiplication.