Scalar expressions : they appear as soon as we differentiate along solutions (see Lyapunov Functions).
Definition
- Motivation: for ,
- which is a quadratic form. What can we say generally about such forms, and in particular, when can they be guaranteed to be negative?
- Quadratic form: any expression like
- where is square, and the same size as
- This defines a scalar, equivalent to
- i.e. the dot product of with
Symmetric and Skew-Symmetric Parts
- Note any square matrix can be written as
- (symmetric)
- In fact
- Then
- Note (a scalar equals its own transpose)
- so
- Hence only the symmetric part of contributes to the quadratic form
- generally assume the matrix in a quadratic form is symmetric
Rayleigh-Ritz Inequality
- Symmetric matrices have real eigenvalues, which leads to
Rayleigh-Ritz Inequality
Definiteness
- A matrix is said to be positive (semi-)definite if
- (by Rayleigh-Ritz)
- is uniformly PD if
- These are the conditions used in the Lyapunov Equation stability theorems