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- TL;DR Summary
- Find minimum and maximum value of quadratic form ##r(\textbf{x})=x_1^2+x_2^2+x_3^3## subject to the constraint ##q(\textbf{x})=x_1^2+3x_2^2+x_3^3+2x_1x_2-2x_1x_3-2x_2x_3=1##.

By working with the following definition of minimum of a quadratic form ##r(\textbf{x})##,

It is clear that the diagonal representation of ##q## is ##y_2^2+4y_3^2##, but how can one apply the above to find the minimum? (Bonus question; does the diagonal representation preserve scale?)

##\lambda_1=\underset{||\textbf{x}||=1}{\text{min}} \ r(\textbf{x})##

where ##\lambda_1## denotes the smallest eigenvalue of ##r##, how would one tackle the above problem?

It is clear that the diagonal representation of ##q## is ##y_2^2+4y_3^2##, but how can one apply the above to find the minimum? (Bonus question; does the diagonal representation preserve scale?)