Pseudo Eigenvalue(Under Construction!)

Pseudo Eigenvalue

http://homepages.neiu.edu/~zzeng/pseudoeig.html

Complex Eigenvalue(Under Construction!)

Complex Eigenvalue

Cauchy eigenvalue interlacing theorem
Gershgorin eigenvalue circle theorem

Spectral Projection

https://www.caam.rice.edu/~caam440/chapter1.pdf

https://en.wikipedia.org/wiki/Resolvent_formalism

(A - zI)     pencil
(A - zI)⁻¹   resolvent / spectra
det(A - zI)  characteristic polynomial
(zI - A)⁻¹ = z⁻¹(I + A/z + A²/z² + ...)
特徵值shift-inverse調到無限大,環積分之後變成特徵向量外積(特徵分解外積表示法)
用C圈選隔離特徵值,得到其特徵向量。
特徵值相異,X是向量;特徵值重根,X是正規正交矩陣。
   -1
∫  --- (A - zI)⁻¹ dz = XXᵀ = P     spectral projection
 C 2πi
         A - λⱼI
Pᵢ = pro -------     Lagrange interpolation
     j≠i λᵢ - λⱼ

專著《Eigenvalues of Matrices》

大量特徵值演算法(Sakurai-Sugiura Projection)

《A projection method for generalized eigenvalue problems using numerical integration》

大量特徵值演算法(FEAST)

《A Density Matrix-based Algorithm for Solving Eigenvalue Problems》

《PFEAST: A High Performance Sparse Eigenvalue Solver Using Distributed-Memory Linear Solvers》

Spectral Projection那一坨積分
用積分演算法求得(Gaussian Quadrature?)
複平面,最大最小特徵值的圓,等距取樣。
自己設定半徑,看要圈多少特徵值。
這樣就得到XXᵀ了。
P = XXᵀ不做內積分解。
而是隨便亂猜X̃,乘在後面(XXᵀ)X̃。
如果猜得準,導致相消剩下X,猜中特徵矩陣。
拿(XXᵀ)X̃來做Rayleigh-Ritz Projection,就這樣。

Quadratic Eigenvalue(Under Construction!)

Quadratic Eigenvalue

https://en.wikipedia.org/wiki/Quadratic_eigenvalue_problem

Generalized Eigenvalue(Under Construction!)

Generalized Eigenvalue

Ax = λBx. if A - zB is singular then z is an eigenvalue.

matrix pencil: A - zB. regular iff det(A - zB) ≠ 0.

B⁻¹A or AB⁻¹ has same eigenvalue.

PₗAPᵣ = λPₗBPᵣx has same eigenvalue.

Weierstrass canonical form:

             [ J-zI |   0  ]
Pₗ(A-zB)Pᵣ = [------+------]
             [   0  | I-zJ ]   below J is nilpotent. diagonal are all 0.

Generalized Schur Decomposition(QZ Decomposition)

A = QSZᵀ
B = QTZᵀ

Biorthogonal Decomposition

biorthogonal decomposition / biorthogonalization

VᵀAW = B    where VᵀW = I

two-side Rayleigh Quotient. generalized eigenvalue. pencil.

VᵀAW
----
 VᵀW

雙正交分解演算法(Unsymmetric Lanczos Iteration)

Lanczos Iteration的時間複雜度實在太銷魂,於是有人硬是用Lanczos Iteration處理一般矩陣。

可以得到三對角線矩陣。