Fundamentals of UD decomposition of KF UD decomposition [1]

It's going on KF UD When decomposing , Variance of state vector - Covariance matrix P Yes, it is UD In the form of ,UD Decomposition is its foundation , The purpose of this article is to reduce the number of UD Decomposed learning time , If it can play the role of throwing bricks and attracting jade, it would be better .
One 、 The purpose of this chapter is three ：
First , Let's be clear , What kind of matrix is a symmetric positive definite matrix ？
secondly , We need to know ,UD The form of decomposition is as follows ：P=UDU^T, among ,U Is a unit upper triangular matrix ,D It's a diagonal matrix .
Last , We need to be able to code , That is, given a symmetric positive definite matrix P, You get the matrix U and D.
The following is lazy operation , Put the screenshot of the document directly ...

stay matlab Chinese handwriting UD The decomposed code is as follows , Please ignore code robustness ：

%  Realize the symmetric positive definite matrix UD decompose 
% demo
% U = 1 2 3 4    D = 1 0 0 0    P = 95 66 38 16
%     0 1 2 3        0 3 0 0        66 47 28 12
%     0 0 1 2        0 0 2 0        38 28 18  8
%     0 0 0 1        0 0 0 4        16 12  8  4

P=[95 66 38 16;
    66 47 28 12;
    38 28 18  8;
    16 12  8  4];

% P=[   220   166   113    66    25
%    166   127    88    52    20
%    113    88    63    38    15
%     66    52    38    24    10
%     25    20    15    10     5];

% UD Principle of decomposition （ From right to left , From bottom to top , List before you go ）
%  because P Is a symmetric positive definite matrix , Only need to P The upper triangular element of 
% 1、 Calculation P Number of matrix columns of 
[pi,pj]=size(P);
U=diag(ones(pi,1));
D=diag(ones(pi,1));
if pj>1
    disp('matrix P is ok!')
    for i=pj:-1:1
        Pii=P(i,i);
        P_D_num=pj-i; %  according to Pii To calculate Dii when , need D Number of elements 
        u2d=0.0;
        if P_D_num>0            
            ip1=i+1;
            for k=ip1:pj
                u2d=u2d+D(k,k)*U(i,k)*U(i,k);
            end            
        end
        D(i,i)=Pii-u2d; % dii
        disp(['P_col= ',num2str(i)])
        
        
        if i==1
            disp('UD over!') %  First column 
        elseif i==pj
            for j=1:pj-1
                U(j,pj)=P(j,pj)/D(i,i); %  The last column 
            end
        else
            ip1=i+1; % plus 1
            im1=i-1; % minus 1
            
            for k=1:im1;
                disp(['P_row= ',num2str(k)])
                udu=0.0;
                for j=ip1:pj
                    udu=udu+U(k,j)*U(i,j)*D(j,j);
                end
                if j==pj
                    U(k,i)=(P(k,i)-udu)/D(i,i);
                end
            end
        end
        disp('-----------')
    end
end
disp('D')
D
disp('U')
U