How to make a matrix positive definite whose elements lie in the range 0.8 to 1 and -0.8 to -1

Question

I have been given a positive definite matrix. Then I have modified the matrix in such a way that all the values of that matrix lie in the range 0.8 to 1 or -0.8 to -1. But after that the matrix no longer remain a positive definite matrix. I have used the

[R,p]=chol(NewMat)

function to verify this. How will I make the modified matrix positive definite?

clc
clf
clear all
close all
Matrix=csvread('new1.csv'); %reading the csv file.
fid1 = fopen('new2.csv');
X = textscan(fid1, '%s%s%s%s%f%f%f', 'Delimiter', ',');
fclose(fid1);
FirstCol = X{1, 1};
SecondCol=X{1,2};
ThirdCol=X{1,3};
FourthCol=X{1,4};
%the factors by which numbers are changed
f1=9.5;
f2=4.75;
f3=3.15;
f4=2.36;
f5=1.9;
f6=1.58;
f7=1.35;
f8=1.18;
flag=1;
p=0;
row=size(Matrix,1);%number of rows in the Matrix
col=size(Matrix,2);%number of columns in the Matrix
NewMat=double(zeros(row,col));
%Factoring the elements of the matrix. The while loop runs until all the
%values lie in the range 0.8<+ve value<1 and -1<-ve value<-0.8

while(flag)
    for i=1:row
        for j=1:col
            if abs(Matrix(i,j))>0 && abs(Matrix(i,j))<=0.1
                NewMat(i,j)=Matrix(i,j)*f1;      
            elseif abs(Matrix(i,j))>0.1 && abs(Matrix(i,j))<=0.2
                NewMat(i,j)=Matrix(i,j)*f2;
            elseif abs(Matrix(i,j))>0.2 && abs(Matrix(i,j))<=0.3
                NewMat(i,j)=Matrix(i,j)*f3;
            elseif abs(Matrix(i,j))>0.3 && abs(Matrix(i,j))<=0.4
                NewMat(i,j)=Matrix(i,j)*f4;
            elseif abs(Matrix(i,j))>0.4 && abs(Matrix(i,j))<=0.5
                NewMat(i,j)=Matrix(i,j)*f5;
            elseif abs(Matrix(i,j))>0.5 && abs(Matrix(i,j))<=0.6
                NewMat(i,j)=Matrix(i,j)*f6;
            elseif abs(Matrix(i,j))>0.6 && abs(Matrix(i,j))<=0.7
                NewMat(i,j)=Matrix(i,j)*f7;
            elseif abs(Matrix(i,j))>0.7 && abs(Matrix(i,j))<=0.8
                NewMat(i,j)=Matrix(i,j)*f8;
            else
                NewMat(i,j)=Matrix(i,j);
            end
        end
    end

    Matrix=NewMat;
    for i=1:row
        for j=1:col
            if (abs(NewMat(i,j))<0.8 || abs(NewMat(i,j))>1)
                flag=1;
                p=1;
            end
        end
    end


    if p==0
    flag=0;
    end
    p=0;
end
%error checking, so that the number remains in the range -1 to +1. If the
%number is not within this range, you can see an error message
for i=1:row
    for j=1:col
        if NewMat(i,j) <-1 || NewMat(i,j) >1
            disp('Error: the number is out of bounds.')
        end
    end
end

Here I have modified the matrix read from the csv file. The modified matrix is NewMat. Its value lie between the above mentioned range, but it is not positive definite. Please help.

Answer 1

The typical question is how do you modify the matrix without altering its eigen values and thus its definiteness. This is typically done with Givens rotations or Housholder reduction. Although these operations are typically discussed in terms of tridiagonalization of a matrix you could likely use them to do other operations after seeing how they are used for tridiagonalization.

If I want to make a n by n matrix positive definite I usually just do something like A=rand(1024,8); A=A'*A; But your question suggests that you want to preserve some unstated property of the original matrix.