Given a vector v in C2, we can define the real part of v (Re(v)) and the imaginary part of v (Im(v)).
Theorem. Let A be a 2×2 matrix with real entries but no real eigenvalues, and let λ be a (complex) eigenvalue of A. The other eigenvalue of A will be the complex conjugate λ of λ.
Proof. AxAx=λx=Ax=Ax=λx=λx
Theorem. Let A be a real 2×2 matrix with complex eigenvalues a−bi(b=0) and v be an associated eigenvector in C2. Then A=PCP−1 where P=[Re(v)Im(v)] and C=(ab−ba).
We know P in this case represents a change of basis; in the equation Ax=PCP−1(x), we first change basis by multiplying by P−1, then undergo some transformation by multiplying by C, and finally change back into standard basis by multiplying by P.
What is the transformation that this matrix C represents?
Thus C can always be represented as the product of a rotation transformation and a scaling transformation.
Inner Products
Rn is a vector space.
You can compute distance between elements of Rn.
Rn is an inner product space.
Definition. An inner product space over R is a vector space V over R combined with an inner product. This is represented as a function ⟨⟩:V×V→R with the following properties: ⟨x,y⟩⟨u+v,w⟩c⟨u,v⟩⟨u,u⟩⟨u,u⟩=⟨y,x⟩=⟨u,w⟩+⟨v,w⟩=⟨cu,v⟩=⟨u,cv⟩≥0=0 iff u=0
Ex. The dot product on Rn is an inner product: ⟨u,v⟩=uTv for u,v in Rn.
Ex. Let C[0,1] be the space of continuous functions on [0,1]. Then ∫01fg is an inner product on C[0,1].
Norm
For any inner product, there is an associated notion of a norm (the magnitude of a vector): ∥v∥=⟨v,v⟩.
For Rn, ⟨v,v⟩∥v∥=v12+...+vn2=v12+...+vn2
With a norm, you also get a notion of distance: dis(u,v)=∥u−v∥.
For Rn, this corresponds to the distance formula.
With an inner product, you also get a notion of what it means for two vectors to be orthogonal to each other: ⟨u,v⟩=0. This is a very important property for vectors, which we will discuss later.