线性代数导论 (6-Eigenvalues and Eigenvectors)
Eigenvalues and Eigenvectors
Introduction to Eigenvalues
The basic equation is $Ax = λx$. The number $λ$ is an eigenvalue of $A$.The vector $x$ is an eigenvector of $A$.
The Equation for the Eigenvalues
$$Ax = λx \
\Downarrow \
(A-λI)x = 0 \
\Downarrow \
det(A-λI)=0
$$
To solve the eigenvalue problem for an n by n matrix, follow these steps:
- Compute the determinant of $A-λI$
- Find the roots of this polynomial
- Solve $(A-λI)x=0$ to find an eigenvector $x$
矩阵A乘以x表示，对向量x进行一次转换（旋转或拉伸）（是一种线性转换），而该转换的效果为常数λ乘以向量x（即只进行拉伸）。
我们通常求特征值和特征向量即为求出该矩阵能使哪些向量（当然是特征向量）只发生拉伸，使其发生拉伸的程度如何（特征值大小）。这样做的意义在于，看清一个矩阵在那些方面能产生最大的效果（power），并根据所产生的每个特征向量（一般研究特征值最大的那几个）进行分类讨论与研究。
Diagonalizing a Matrix
Suppose the $n$ by $n$ matrix $A$ has $n$ linearly independent eigenvectors $x_1,\ldots,x_n$.(Without n independent eigenvectors, we can’t diagonalize.) Put them into the columns of an eigenvector matrix $S$. Eigenvalue matrix $Λ$
$$
Λ=\begin{bmatrix} λ_1 \ & \ddots \ & & λ_n \end{bmatrix}
$$
than
$$A=SΛS^{-1}$$
Remember that there is no connection between invertibility and diagonalizability:
Invertibility is concerned with the eigenvalues ($λ = 0$ or $λ \neq 0$).
Diagonalizability is concerned with the eigenvectors (too few or enough for $S$).
If that diagonal matrix has any zeroes on the diagonal, then $A$ is not invertible. Otherwise, $A$ is invertible.
$\begin{aligned} \textbf{Powers of }A \end{aligned} \qquad A^k =SΛ^kS^{-1}$
Applications to Differential Equations
Symmetric Matrices
What is special about $Ax = λx$ when A is symmetric?
- A symmetric matrix has only real eigenvalues.
- The eigenvectors can be chosen orthonormal.
Why do we use the word “choose”? Because the eigenvectors do not have to be unit vectors. Their lengths are at our disposal. We will choose unit vectors-eigenvectors of length one, which are orthonormal and not just orthogonal. Then $SΛS^{-1}$ is in its special and particular form $QΛQ^T$ for symmetric matrices.
Symmetric diagonalization : Every symmetric matrix has the factorization $A=QΛQ^T=QΛQ^{-1}$ with real eigenvalues in $Λ$ and orthonormal eigenvectors in $S=Q$.
All Symmetric Matrices are Diagonalizable
Every square matrix can be “triangularized” by $A = QTQ^{-1}$.
Positive Definite Matrices
Symmetric matrices that have positive eigenvalues are called positive definite.
正定矩阵等价判定条件:
- All $n$ eigenvalues are positive
- All $n$ pivots are positive
- All $n$ upper left determinants are positive
- $x^TAx$ is positive except at $x=0$.(This is the energy-based definition.)
- $A$ equals $R^TR$ for a matrix R with independent columns.
$A = R^TR$ is automatically positive definite if $R$ has independent columns.
Similar Matrices
Diagonalization is not possible for every $A$. Some matrices have too few eigenvectors. In this new section, the eigenvector matrix $S$ remains the best choice when we can find it, but now we allow any invertible matrix $M$.
A typical matrix $A$ is similar to a whole family of other matrices because there are so many choices of $M$.
Let $M$ be any invertible matrix. Then $B = M^{-1}AM$ is similar to $A$.
Singular Value Decomposition (SVD)
The eigenvectors in $S$ have three big problems: They are usually not orthogonal, there are not always enough eigenvectors, and $Ax = λx$ requires $A$ to be square. The singular vectors of $A$ solve all those problems in a perfect way.
$$A = UΣV^T$$
The $u$’s are eigenvectors of $AA^T$ and the $v$’s are eigenvectors of $A^TA$.
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