“This is the 12th day of my participation in the First Challenge 2022. For details: First Challenge 2022.”

@TOC

preface

Hello! Friend!!! ଘ(੭, ᵕ)੭ Nickname: Haihong Name: program monkey | C++ player | Student profile: Because of C language, I got acquainted with programming, and then transferred to the computer major, and had the honor to win some state awards, provincial awards… Has been confirmed. Currently learning C++/Linux/Python learning experience: solid foundation + more notes + more code + more thinking + learn English! Machine learning small white stage article only as their own learning notes for the establishment of knowledge system and review know why!

The articles

Matrix theory for Machine Learning (1) : Sets and Mappings

Matrix Theory for Machine Learning (2) : Definitions and Properties of linear Spaces

Matrix theory (3) : Bases and coordinates of linear Spaces

Matrix theory for Machine Learning (4) : Basis transformation and coordinate transformation

Matrix theory for Machine Learning (5) : Linear subspaces

Matrix theory (6) : Intersection and sum of subspaces

Matrix theory for Machine Learning (7) : Euclidean Spaces

Matrix theory (8) : Orthonormal basis and Gram-Schmidt Process

Matrix theory for Machine Learning (9) : Orthogonal complement and projection theorem

Matrix theory for Machine Learning (10) : Definition of linear Transformations

Matrix theory (11) : Matrix representation of linear transformations

Matrix theory for Machine Learning (12) : Approximation theory

Matrix theory for Machine Learning (13) : Hamliton-Cayley theorem, minimum polynomials

Matrix theory for Machine Learning (14) : Vector norm and Its Properties

Matrix theory (15) : The norm of matrices

5.1 Limits of vectors and Matrices

5.1.1 Vector sequence limits

Definition 5.1

Set a given NNN dimensional vector space CnC ^ nCn in vector sequence {χ (k)} \ {\ boldsymbol \ chi ^ {(k)} \} {χ (k)}, among them

$\boldsymbol\chi^{(k)}=(\xi_1^{(k)},\xi_2^{(k)},… , \ xi_n ^ {(k)}) \ quad k = 1, 2, 3,…$

If each component of the factor (k) I had \ xi_i ^ {} (k) factor (k) when k I had – up k \ rightarrow \ inftyk – up, has a limit factor I had \ xi_i factor I, namely

$\ \ lim_ {k rightarrow \ infty} \ xi_i ^ = {(k)} \ xi_i \ quad I = 1, 2,… ,n$

Remember χ = (factor 1 and factor 2,… , factor n) \ boldsymbol \ chi = (\ xi_1 \ xi_2,… , \ xi_n) χ = (factor 1 and factor 2,… , factor n)

Says vector sequence {χ (k)} \ {\ boldsymbol \ chi ^ {(k)} \} {χ (k)} have limit or {χ (k)} \ {\ boldsymbol \ chi ^ {(k)} \} {χ (k)} converges to \ boldsymbol \ chi χ

For short {χ(k)}\{boldsymbol\chi^{(k)}\}{χ(k)} convergence

$\ lim_ \ rightarrow \ infty {k} \ boldsymbol \ chi ^ {} (k) = \ boldsymbol \ chi or \ boldsymbol \ chi ^ {(k)} \ rightarrow \ boldsymbol \ chi$

Vector sequence {χ (k)} \ {\ boldsymbol \ chi ^ {(k)} \} {χ (k)} contains a lot of items Simple to understand: the χ (1), χ (2), χ (3)… χ (k) \ boldsymbol \ chi ^ {(1)}, \ boldsymbol \ chi ^ {(2)}, \ boldsymbol \ chi ^ {} (3)… \boldsymbol\chi^{(k)}χ(1), χ(2), χ(3)… With the increase of KKK, the composition of χ(K) gradually approaches the vector χ\ BoldSymbol \ Chi χ

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Defined by the vector sequence limit, vector sequence {χ (k)} \ {\ boldsymbol \ chi ^ {(k)} \} {χ (k)} converges to χ \ boldsymbol \ chi the sufficient and necessary conditions

• {χ (k) – χ} \ {\ boldsymbol \ chi ^ {} (k) – \ boldsymbol \ chi \} {χ (k) – χ} converges to zero vector

Or is it

• A vector norm for arbitrary ∣ ∣ ⋅ ∣ ∣ | | \ cdot | | ∣ ∣ ⋅ ∣ ∣, sequence ∣ ∣ {χ (k) – χ} ∣ ∣ | | \ {\ boldsymbol \ chi ^ {} (k) – \ boldsymbol \ chi \} | | ∣ ∣ {χ (k) – χ} ∣ ∣ converges to zero

5.1.2 Square matrix sequence limit

Definition 5.2

For the compound matrix sequence {Ak}\{A_k\}{Ak}, where

If in the k – up k \ rightarrow \ inftyk – up, n2n ^ 2 n2 complex sequence {aij (k)} \ {a_ {ij} ^ {(k)} \} {aij (k)} are separately converge to aija_ {ij} aij, namely

$A_ \ rightarrow \ infty \ lim_ {k} {the ij} ^ {} (k) = a_ {ij} \ quad (I, j = 1, 2,… ,n)$

Then AAA is called the limit of {Ak}\{A_k\}{Ak} at k→∞k\rightarrow\inftyk→∞, denoted as

$\lim_{k\rightarrow\infty}A_k=A$

namely

若
$k\rightarrow\infty$When,
$\{A_k\}$If it does not converge, it is called square matrix sequence
$\{A_k\}$Is divergent

As long as the Ak} {\ {A_k \} {Ak} one element aij (k) a_ ^ {ij} {(k)} aij (k) in the k – up k \ rightarrow \ inftyk – up not convergence, The Ak} {\ {A_k \} {Ak} must be no convergence Only the Ak} {\ {A_k \} {Ak} all elements aij (k) a_ ^ {ij} {(k)} aij (k) in the k – up k \ rightarrow \ inftyk – up convergence, Then {Ak}\{A_k\}{Ak} converges

Theorem 5.1.1

The sufficient and necessary conditions for the matrix sequence {Ak}\{A_k\}{Ak} in Cn×nC^{n×n}Cn×n converges to square matrix AAA are: A square matrix norm for arbitrary ∣ ∣ ⋅ ∣ ∣ | | \ cdot | | ∣ ∣ ⋅ ∣ ∣, sequence {∣ ∣ Ak – A ∣ ∣} \ {| | A_k -a | | \} {∣ ∣ Ak – A ∣ ∣} converges to zero

{Ak}\{A_k\}{Ak} is a sequence of matrices, specifically A1,A2,A3… ,AkA_1,A_2,A_3,… ,A_kA1,A2,A3,… , Ak {∣ ∣ Ak ∣ ∣} \ {| | A_k | | \} {∣ ∣ Ak ∣ ∣} is the general series, concrete is ∣ ∣ A1 ∣ ∣, ∣ ∣ A2 ∣ ∣,… , ∣ ∣ Ak ∣ ∣ | | A_1 | | to | | A_2 | |,… , | | A_k | | ∣ ∣ A1 ∣ ∣, ∣ ∣ A2 ∣ ∣,… , ∣ ∣ Ak ∣ ∣ because ∣ ∣ Ak ∣ ∣ | | A_k | | ∣ ∣ Ak ∣ ∣ said only a, so {∣ ∣ Ak ∣ ∣} \ {| | A_k | | \} {∣ ∣ Ak ∣ ∣} is a regular sequence

$C ^ {n * n}$Properties of convergent square matrix sequences in

– up Ak (1) if the lim ⁡ k = A \ lim_ \ rightarrow \ infty} {k A_k = Alimk – up Ak = A, for Cn x nC ^ {n * n} Cn x n in any square matrix norm ∣ ∣ ⋅ ∣ ∣ | | \ cdot | | ∣ ∣ ⋅ ∣ ∣, ∣ ∣ Ak ∣ ∣ | | A_k | | ∣ ∣ Ak ∣ ∣ bounded

(2) If lim⁡k→∞Ak=A,lim⁡k→∞Bk=B\lim_{k\rightarrow\infty}A_k=A,\lim_{k\rightarrow\infty}B_k=Blimk→∞Ak=A,limk→∞Bk=B, And lim – up ak ⁡ k = a, lim ⁡ k – up bk = b \ lim_ \ rightarrow \ infty} {k a_k = a, \ lim_ \ rightarrow \ infty} {k b_k = blimk – up ak = a, limk – up bk = b, {ak},{bk}\{a_k\},\{b_k\}{ak},{bk} is a sequence, then

$\lim_{k\rightarrow\infty}(a_kA_k+b_kB_k)=aA+bB$

(3) If lim⁡k→∞Ak=A,lim⁡k→∞Bk=B\lim_{k\rightarrow\infty}A_k=A,\lim_{k\rightarrow\infty}B_k=Blimk→∞Ak=A,limk→∞Bk=B, then

$\lim_{k\rightarrow\infty}A_kB_k=\lim_{k\rightarrow\infty}A_k\cdot\lim_{k\rightarrow\infty}B_k=AB$

(4) lim – up Ak ⁡ k = A \ lim_ \ rightarrow \ infty} {k A_k = Alimk – up Ak = A, and Ak – 1 A_k ^ {1} Ak – 1-1 A and A ^ {1} – 1 A, then

$\lim_{k\rightarrow\infty}A_k^{-1}=A^{-1}$

Jordan block to a positive integer power

set

是
$t$Order Jordan block, and remember
$t$Order matrix

There are

$J(\lambda_0,t)=\lambda_0 E + H\quad(E is t order identity matrix)$

Due to the

及
$H^t=0$, there is

$J^k(\lambda_0,t)=(\lambda_0 E + H)^k$

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Similarly, for numerical variable XXX, by

$J(\lambda_0,t)x=\lambda_0xE+xH$

available

Theorem 5.1.2

A∈Cn×nA\in C^{n×n}A∈Cn×n power E,A,A2… ,Ak,… E,A,A^2,… ,A^k,… E,A,A2,… ,Ak,… The resultant sequence of matrices {Ak}\{A^k\}{Ak} converges to zero matrices only if the modulo of the eigenvalues of AAA are all less than 1

proposition

Set ∣ ∣ A ∣ ∣ A | | A | | _a ∣ ∣ A ∣ ∣ A is compatible with vector norm ∣ ∣ χ ∣ ∣ A | | \ boldsymbol \ chi | | _a ∣ ∣ χ ∣ ∣ A phalanx of norm, then rho (A) or less ∣ ∣ A ∣ ∣ \ A rho (A) \ leq | | A | | _a rho (A) or less ∣ ∣ A ∣ ∣ A

Theorem 5.1.3

Ak – > 0 A ^ {k} \ rightarrow0Ak – > 0, the sufficient and necessary condition that there are at least A square matrix norm ∣ ∣ ⋅ ∣ ∣ | | \ cdot | | ∣ ∣ ⋅ ∣ ∣, make ∣ ∣ A ∣ ∣ < 1 | | A | | < 1 ∣ ∣ A ∣ ∣ < 1

conclusion

Description:

• Refer to matrix Theory in your textbook
• With the book concept explanation combined with some of their own understanding and thinking

The essay is just a study note, recording a process from 0 to 1

Hope to have a little help to you, if there is a mistake welcome small partners correct