MATH 240

Fri. November 15th, 2019

Given a vector vv in C2\cnums^2, we can define the real part of vv (Re(v)\text{Re}(v)) and the imaginary part of vv (Im(v)\text{Im}(v)).

Theorem. Let AA be a 2×22\times 2 matrix with real entries but no real eigenvalues, and let λ\lambda be a (complex) eigenvalue of AA. The other eigenvalue of AA will be the complex conjugate λ‾\overline{\lambda} of λ\lambda.

Proof.
Ax=λxAx‾=A‾x‾=Ax‾=λx‾=λ‾x‾\begin{aligned} Ax&=\lambda x\\ A\overline{x}&=\overline{A}\overline{x}\\ &=\overline{Ax}\\ &=\overline{\lambda x}\\ &=\overline{\lambda}\overline{x} \end{aligned}

Theorem. Let AA be a real 2×22\times 2 matrix with complex eigenvalues a−bi (b≠0)a-bi\ (b\neq 0) and vv be an associated eigenvector in C2\cnums^2. Then A=PCP−1A=PCP^{-1} where P=[Re(v)Im(v)]P=\begin{bmatrix} \text{Re}(v)&\text{Im}(v) \end{bmatrix} and C=(a−bba)C=\begin{pmatrix} a&-b\\b&a \end{pmatrix}.

We know PP in this case represents a change of basis; in the equation Ax=PCP−1(x)Ax=PCP^{-1}(x), we first change basis by multiplying by P−1P^{-1}, then undergo some transformation by multiplying by CC, and finally change back into standard basis by multiplying by PP.

What is the transformation that this matrix CC represents?

C=(a−bba)det(C−λI)=∣a−λ−bba−λ∣=(λ−a)2+b20=(λ−a)2+b2λ=a±bir=∣λ∣=a2+b2C=r(aa2+b2−ba2+b2ba2+b2aa2+b2)=r(cos⁡θ−sin⁡θsin⁡θcos⁡θ)=(r00r)(cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{aligned} C&=\begin{pmatrix} a&-b\\b&a \end{pmatrix}\\ \text{det}(C-\lambda I)&=\begin{vmatrix} a-\lambda&-b\\b&a-\lambda \end{vmatrix}\\ &=(\lambda -a)^2+b^2\\ 0&=(\lambda -a)^2+b^2\\ \lambda&=a\pm bi\\ r&=\mid\lambda\mid=\sqrt{a^2+b^2}\\ C&=r\begin{pmatrix} \dfrac{a}{\sqrt{a^2+b^2}}&\dfrac{-b}{\sqrt{a^2+b^2}}\\\dfrac{b}{\sqrt{a^2+b^2}}&\dfrac{a}{\sqrt{a^2+b^2}} \end{pmatrix}\\ &=r\begin{pmatrix} \cos{\theta}&-\sin{\theta}\\\sin{\theta}&\cos{\theta} \end{pmatrix}\\ &=\begin{pmatrix} r&0\\0&r \end{pmatrix}\begin{pmatrix} \cos{\theta}&-\sin{\theta}\\\sin{\theta}&\cos{\theta} \end{pmatrix} \end{aligned}

Thus CC can always be represented as the product of a rotation transformation and a scaling transformation.

Inner Products

  1. Rn\R^n is a vector space.
  2. You can compute distance between elements of Rn\R^n.
  3. Rn\R^n is an inner product space.

Definition. An inner product space over R\R is a vector space VV over R\R combined with an inner product. This is represented as a function ⟨⟩:V×V→R\langle \rangle:V\times V\rightarrow \R with the following properties:
⟨x,y⟩=⟨y,x⟩⟨u+v,w⟩=⟨u,w⟩+⟨v,w⟩c⟨u,v⟩=⟨cu,v⟩=⟨u,cv⟩⟨u,u⟩≥0⟨u,u⟩=0 iff u=0\begin{aligned} \langle x,y\rangle&=\langle y,x\rangle\\ \langle u+v,w\rangle&=\langle u,w\rangle+\langle v,w\rangle\\ c\langle u,v\rangle&=\langle cu,v\rangle=\langle u,cv\rangle\\ \langle u,u\rangle&\geq0\\ \langle u,u\rangle&=0 \text{ iff } u=0 \end{aligned}

Ex. The dot product on Rn\R^n is an inner product: ⟨u,v⟩=uTv\langle u,v\rangle=u^Tv for u,vu,v in Rn\R^n.

Ex. Let C[0,1]C[0,1] be the space of continuous functions on [0,1][0,1]. Then ∫01fg\int_0^1 fg is an inner product on C[0,1]C[0,1].

Norm

For any inner product, there is an associated notion of a norm (the magnitude of a vector): ∥v∥=⟨v,v⟩\lVert v\rVert=\sqrt{\langle v,v\rangle}.

For Rn\R^n,
⟨v,v⟩=v12+...+vn2∥v∥=v12+...+vn2\begin{aligned} \langle v,v\rangle&=v_1^2+...+v_n^2\\ \lVert v\rVert&=\sqrt{v_1^2+...+v_n^2} \end{aligned}

With a norm, you also get a notion of distance: dis(u,v)=∥u−v∥\text{dis}(u,v)=\lVert u-v\rVert.
For Rn\R^n, 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\langle u,v\rangle=0. This is a very important property for vectors, which we will discuss later.