Check Positive Definite Matrix in Matlab

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It is often required to check if a given matrix is positive definite or not. Three methods to check the positive definiteness of a matrix were discussed in a previous article .

I will utilize the test method 2 to implement a small matlab code to check if a matrix is positive definite.The test method 2 relies on the fact that for a positive definite matrix, the determinants of all upper-left sub-matrices are positive.The following Matlab code uses an inbuilt Matlab function -‘det’ – which gives the determinant of an input matrix.

Furthermore, the successive upper \(k \times k\) sub-matrices are got by using the following notation

subA=A(1:i,1:i)

I will explain how this notation works to give the required sub-matrices.

Consider a sample matrix given below

$$ \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}$$

The sub-matrices for the various combinations for row and column values for the above mentioned code snippet is given below

Notation Submatrix Comment
subA=A(1:1,1:1) \( \begin{bmatrix} 1 \end{bmatrix}\) Select elements from 1st row-1st column to 1st row-1st column
subA=A(1:2,1:2) \( \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}\) Select elements from 1st row-1st column to 2nd row-2nd column
subA=A(1:3,1:3) \( \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}\) Select elements from 1st row-1st column to 3rd row-3rd column
subA=A(1:3,1:2) \( \begin{bmatrix} 1 & 2 \\ 4 & 5 \\ 7 & 8 \end{bmatrix}\) Select elements from 1st row-1st column to 3rd row-2nd column

Matlab Code to test if a matrix is positive definite:

function x=isPositiveDefinite(A)
%Function to check whether a given matrix A is positive definite
%Author Mathuranathan for https://www.gaussianwaves.com
%Returns x=1, if the input matrix is positive definite
%Returns x=0, if the input matrix is not positive definite
%Throws error if the input matrix is not symmetric

    %Check if the matrix is symmetric
    [m,n]=size(A); 
    if m~=n,
        error('A is not Symmetric');
    end
    
    %Test for positive definiteness
    x=1; %Flag to check for positiveness
    for i=1:m
        subA=A(1:i,1:i); %Extract upper left kxk submatrix
        if(det(subA)<=0); %Check if the determinent of the kxk submatrix is +ve
            x=0;
            break;
        end
    end
    
    if x
        display('Given Matrix is Positive definite');
    else
        display('Given Matrix is NOT positive definite');
    end      
end

Sample run:

>> A=[1 2 3; 4 5 6; 7 8 9]
\(A =\begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9\end{bmatrix}\)
>> x=isPositiveDefinite(A)
Given Matrix is NOT positive definite
x = 0
------------------------------------------
>> A=[25 15 -5; 15 18 0;-5 0 11]
\(A =\begin{bmatrix}
25 & 15 & -5\\
15 & 18 & 0\\
-5 & 0 & 11 \end{bmatrix}\)
>> x=isPositiveDefinite(A)
Given Matrix is Positive definite
x = 1
------------------------------------------
>> A=[1 2 3; 4 5 6]
\(A =\begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6 \end{bmatrix}\)
>> x=isPositiveDefinite(A)
Error using isPositiveDefinite (line 11)
A is not Symmetric
------------------------------------------

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