Linear Algebra and Tensors

Matrices, linear transformations, eigenvectors, tensors, and normal modes in physics.

General idea of the topic

Linear algebra provides the mathematical backbone for multivariable physical systems, relativistic coordinate transformations, and quantum mechanics formulated on infinite-dimensional Hilbert spaces. It formalizes how physical observables transform under spatial rotations, Lorentz boosts, and curvilinear coordinate changes, categorizing geometric invariants into scalars (rank 0), contravariant/covariant vectors (rank 1), and general tensors (rank $\ge 2$).

This node covers vector spaces, linear operator transformations, eigenvalue/eigenvector spectral decomposition, Hermitian matrix diagonalization, and tensor calculus in Minkowski and curved spacetimes.

What you will truly understand

You will understand how diagonalizing a symmetric tensor matrix (such as the rotational moment of inertia tensor $I_{ij}$ or Cauchy stress tensor $\sigma_{ij}$) identifies the orthogonal principal axes where shear components vanish and physics uncouples.

You will master Einstein summation convention ($A_i B^i$) and index gymnastics (raising and lowering indices $v_\mu = g_{\mu\nu} v^\nu$) in non-Euclidean metric manifolds.

Recommended learning route

1. Vector spaces and basis sets: linear independence, orthonormal bases, inner product spaces, and Gram-Schmidt orthogonalization.
2. Matrices and linear transformations: determinants, trace invariants, orthogonal rotation groups $SO(3)$, and unitary groups $U(N)$.
3. Spectral decomposition and eigenvalues: characteristic polynomial $\det(A - \lambda I) = 0$, Hermitian spectral theorem, and coupled normal vibrational modes.
4. Tensor analysis and metric spaces: transformation laws, metric tensor $g_{\mu\nu}$, index contraction, and Levi-Civita permutation symbol $\epsilon_{ijk}$.

Physical criterion you must acquire

The principle of general covariance: a physical law expressed in tensor equation form (such as $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$ or $G_{\mu\nu} = \kappa T_{\mu\nu}$) retains identical mathematical validity across all coordinate reference frames.

Connection with real problems

• Rigid body dynamics: 3D inertia tensors, principal axes of rotation, and Euler angles.
• Quantum mechanics: Hermitian operators as measurable physical observables and matrix mechanics.
• Special and General Relativity: Minkowski metric signature $\eta_{\mu\nu}$ and spacetime curvature tensors.

Final goal of the topic

Master matrix algebra, spectral eigenvalue decomposition, and covariant tensor calculus for advanced classical and quantum physics.