Linear Algebra III: Spectral Theory, SVD & Applications
A capstone third linear algebra course covering the spectral theorem, SVD, conditioning and data-science applications such as PCA and Markov chains.
What you’ll learn
- Use the spectral theorem for symmetric matrices and quadratic forms
- Understand the singular value decomposition as rotate–stretch–rotate
- Apply SVD and PCA to compress and denoise real datasets
- Reason about conditioning, stability and iterative methods
Curriculum
M1
Spectral theorem & quadratic forms — 4 lessons
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Symmetric matrices & orthogonal diagonalization16m
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Spectral theorem (real symmetric case)16m
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Quadratic forms & classification16m
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Rayleigh quotient & extremal eigenvalues14m
M2
SVD & data applications (PCA) — 3 lessons
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SVD: rotate–stretch–rotate view16m
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Low-rank approximations & energy16m
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PCA via SVD: data compression & denoising18m
M3
Conditioning, numerics & iterative methods — 3 lessons
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Matrix norms & condition numbers16m
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Ill-conditioning & floating-point issues16m
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Iterative methods: power method & basic solvers16m
M4
Applied projects & case studies — 3 lessons
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Markov chains & steady states16m
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Linear systems of ODEs & eigen decomposition16m
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Mini-project: from data to model using LA tools18m
Projects
- Diagonalise symmetric matrices and interpret quadratic forms geometrically
- Run PCA on a real dataset and visualise principal components
- Implement the power method and explore convergence in practice
Prerequisites
- Solid Linear Algebra II background (eigenvalues, inner products, projections)
- Comfort with basic probability and calculus is helpful for applications
- Some coding experience (Python/NumPy) strongly recommended
Who is this for?
- Students in an advanced Linear Algebra or Applied Linear Algebra course
- Data science / ML learners who want theory behind PCA and spectral methods
- Anyone heading into numerical analysis, optimisation or quantum mechanics
Outcomes
- Able to connect SVD, eigen decompositions and quadratic forms in one picture
- Comfortable using PCA and related tools on real datasets
- Ready to tackle numerical linear algebra, optimisation and advanced ML
Resources
- Starter project / template (ZIP)
- Setup & study checklists (PDF)
- Core formulas / syntax cheatsheet (PDF)
Tip: Right-click → “Save link as…” if your browser opens the file.
FAQ
Is this beginner-friendly?
Yes. We start from mental models and build up with guided practice and plenty of worked examples.
Will I need extra software?
We stick to free or standard tools. Any optional extras are clearly marked and have alternatives.
Do I get updates?
Yes—lifetime access with updates as the field, tools, and exam expectations evolve.