Linear Algebra › Section G · Symmetric matrices and quadratic forms › Topic G.3
Constrained optimization
How large can a quadratic form get when its input is restricted to unit vectors, and in which direction is that maximum reached? The answer is an eigenvalue and its eigenvector, and the same idea locates the greatest stress in a beam, the direction of greatest variance in data and the greatest stretch of any matrix.
In this topic
- Why a constraint is needed
- The diagonal case
- The theorem for any symmetric matrix
- Worked example: a form with a cross term
- The next largest value
- Worked example: three variables, step by step
- The Rayleigh quotient and Lagrange multipliers
- Worked example: principal stresses
- Common mistakes to avoid
- Key takeaways
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