Note of Michael Artin "Algebra" Chapter 5 "Applications of Linear Operators" (to complete)

题记

说实话这一部分看起来和抽象代数关系不大。但是这一部分涉及了一元向量函数和“矩阵函数”的连续性、求导和常微分方程的计算。常微分方程北航的数学分析使用的是北大伍胜健的教材，是不教微分方程的。另外有刚刚接触的群论在正交变换上的应用和数形结合，因此值得一看。

5.1 ORTHOGONAL MATRICES AND ROTATIONS

5.1.1 Def. (Dot Product) (Skipped)

• The length of a vector \(X\) is \((X\cdot X) = X^TX\).
• A vector \(X\) is orthogonal to another vector \(Y\), written \(X \perp Y\) if and only if \(X^TY = 0\).

5.1.5 Theo. (Pythagoras)

If \(X\perp Y\) and \(Z = X+Y\), then \(|Z|^2=|X|^2 + |Y|^2\).

5.1.7 Lemma.

Any set \((v_1,\cdots,v_n)\) of orthogonal nonzero vectors in \(\mathbb{R}^n\) is independent.

5.1.9 Def.

A real \(n\times n\) matrix \(A\) is orthogonal if \(A^TA=I\), which is to say, \(A\) is invertible and its inverse is \(A^T\).

5.1.10 Lemma.

An \(n\times n\) matrix \(A\) is orthogonal if and only if ites columns form an orthogonal basis of \(\mathbb{R}^n\).

5.1.11 Prop.

• The product, inverse, transpose of orthogonal matrices is orthogonal. The orthogonal matrices form a subgroup \(O_n\) of \(GL_n\), the orhogonal group.
• \(\det (o) = \pm 1, o \in O_n\). The orthogonal matrices with determinant \(1\) form a subgroup \(SO_n\) of \(O_n\) of index \(2\), the special orthogonal group.

5.1.12 Def. (Orthogonal Operator)

An orthogonal operator \(T\) on \(\mathbb{R}^n\) is a linear operator that preserves the dot product:

\[(TX \cdot TY) = (X\cdot Y), \forall X,Y \in \mathbb{R}^n \]

5.1.13 Prop.

A linear operator \(T\) on \(\mathbb{R}^n\) is orthogonal if and only if \(\forall X, (TX\cdot TX) = (X\cdot X)\).

5.1.14 Prop.

A linear operator \(T\) on \(\mathbb{R}^n\) is orthogonal if and only if its matrix \(A\) with respect to the standard basis is an orthogonal matrix.

5.1.15 Lemma.

Let \(M\) be an \(n\times n\) matrix, If \(X^TMY=0\) for all column vectors \(X\) and \(Y\), then \(M=0\).

5.1.17 Theo.

• (a) The orthogonal \(2\times 2\) matrices with determinant \(1\) are the matrices

\[R = \begin{bmatrix} c & -s\\ s & c \end{bmatrix} \]

• (b) The orthogonal \(2\times 2\) matrices with determinant \(-1\) are the matrices

\[S = \begin{bmatrix} c & s\\ s & -c \end{bmatrix} = RS_0 \]

• The matrix \(S\) reflects the plane about the one-dimensional subspace of \(\mathbb{R}^2\) that makes an angle \(\frac{1}{2}\theta\) with the \(e_1\)-axis.

5.1.22 Def. (Rotation of \(\mathbb{R}^3\))

A rotation of \(\mathbb{R}^3\) about the origin is a linear operator \(\rho\) with these properties:

• \(\rho\) fixes a unit vector \(u\), called a pole of \(\rho\), and
• \(\rho\) rotates the two-dimensional subspace \(W\) orthogonal to \(u\).

The angle \(\theta\) perpendicular to the pole \(u\) follows the "Right Hand Rule".
When \(u\) is the \(e_1\), the set \((e_2,e_3)\) will be a basis for \(W\), and the matrix of \(\rho\) woll have the form

\[M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & c & -s \\ 0 & s & c \end{bmatrix} \]

A rotation is not the identity is described bt the pair \((u, \theta)\), called a spin, that consosts of a pole \(u\) and a nonzero angle of rotation \(\theta\).

5.1.25 Theo. (Euler's Theorem)

The \(3\times 3\) rotation matrices are the orthogonal \(3\times 3\) matrices with determinant \(1\), the elements of the special orthogonal group \(SO_3\) (three-dimensional rotation groups).

5.1.26 Cor.

The composition of rotations about any two axes is a rotation about some other axis.

5.1.28 Cor.

Let \(M\) be the matrix in \(SO_3\) that represents the rotation \(\rho_{(u,\alpha)}\) with spin \((u,\alpha)\).

• (a) The trace of \(M\) is \(1+2\cos \alpha\).
  • PS: The matrices are communicative in trace.
• (b) Let \(B \in SO_3\), and let \(u' = Bu\), The conjugate \(M' = BMB^T\) represents the rotation \(\rho(u', \alpha)\) with spin \((u', \alpha)\).

5.1.29 Lemma.

A \(3\times 3\) orthogonal matrix \(M\) with determinant \(1\) has an eigenvalue equal to \(1\).

5.2 USING CONTINUITY

5.2.1 Prop. (Continuity of Roots)

Let \(p_k(t)\) be a sequence of monic polynomials of degree \(\leq n\), and let \(p(t)\) be another monic polynomial of degree \(n\). Let \(\alpha_{k,1},\cdots ,\alpha_{k,n}\) and \(a_1,\cdots ,a_n\) denote the roots of these polynomials.

• (a) If \(\alpha_{k,v} \to \alpha_v\) for \(v = 1,\cdots ,n\), then \(p_k\to p\).
• (b) Conversely, if \(p_k\to p\), the roots \(\alpha_{k,v}\) of \(p_k\) can be numbered in such a way that \(\alpha_{k,v}\to \alpha_v\) for each \(v=1,\cdots ,n\).

5.2.2 Prop.

Let \(A\) be an \(n\times n\) complex matrix.

• (a) There is a sequance of matrix \(A_k\) that converges to \(A\), and such that for all \(k\) the characteristic polynomial \(p_k(t)\) of \(A\).
• (b) If a subsequence \(A_k\) of matrices converges to \(A\), the sequence \(p_k(t)\) of its characteristic polynomials converges to the characteristic polynomial \(p(t)\) of \(A\).
• (c) Let \(\lambda_i\) be the roots of the characteristic polynomial \(p\). If \(A_k \to A\), the roots \(\lambda_{k,i}\) of \(p_k\) can be numbered so that \(\lambda_{k,i} \to \lambda_i\) for each \(i\).

5.2.3 Theo. (Cayley-Hamilton Theorem) (Skipped)

特征多项式代入原矩阵后是零化多项式。

5.3 SYSTEMS OF DIFFERENTIAL EQUATIONS

这一张是单元向量函数和矩阵函数的常微分方程的有解性，并给出了矩阵可对角化时求解方式。

5.3.7

A system of homogeneous linear, first-order, contant coefficient differential equations is a matrix equation of the form

\[\frac{\mathrm{d}X}{\mathrm{d}t}=AX \]

5.3.11~5.3.13

If \(A = \mathrm{diag}(\lambda_1, \cdots, \lambda_n)\), we have

\[x_i = c_ie^{\lambda_i t} \]

Therefore, if \(V\) is an eigenvector for \(A\) with eigenvalue \(\lambda\), i.e., if \(AV=\lambda V\), then

\[X=e^{\lambda t}V \]

is a particular solution of (5.3.7). Here it is interpreted as the product of the variable scalar \(e^{\lambda t}\) and the constant vector \(V\).

\[\frac{\mathrm{d}}{\mathrm{d}t} e^{\lambda t} V = \lambda e^{\lambda t}V = A e^{\lambda t} V \]

This observation allows us ti solve (5.3.7) whenever the matrix \(A\) has distinct real eigenvalues. In that case every solution will be a linear combination of the special solutions (5.3.13).
To work out this, it is convenient to diagonalize.

5.3.15 Prop.

Let \(A_{n\times n}\) be a matrix, and \(P\) be an invertible matrix such that \(\Lambda = P^{-1}AP = \mathrm{diag} (\lambda_1, \cdots, \lambda_n)\). The general solution to \(\frac{\mathrm{d}X}{\mathrm{d}t} =AX\) is \(X=P\widetilde{X}\), where \(\widetilde{X}=(c_1 e^{\lambda_1 t}, \cdots, c_n e^{\lambda_n t})^T\) solves the equation \(\frac{\mathrm{d}\widetilde{X}}{\mathrm{d}t} =\Lambda \widetilde{X}\).