Sylvester's criterion

The Sylvester’s criterion can be used to check whether a symmetric matrix is positive definite

A quadratic form ( symmetric matrix) is positive definite if the leading principal minors of are positive. Leading principal minors are the determinants of the top-left square submatrices of a given matrix Image credit: Princeton lecture

Code

Sagemath

def is_positive_definite(A):
    """Apply Sylvester's Criterion to check if a matrix is positive definite. 
    
    Inputs:
    - A: A symmetric matrix
    
    Output:
    - True if the matrix is positive definite, False otherwise
    """
    # Ensure matrix is symmetric
    if A.is_symmetric():
	    print(f"Matrix is not symmetric")
	    return False
    
    n = A.nrows()
    
    # Loop through submatrices
    for k in range(1, n+1):
        # Extract the top-left k x k submatrix
        submatrix = A[:k, :k]
        determinant = submatrix.det()
        
        # If any determinant is non-positive, the matrix is not positive definite
        if determinant <= 0:
            print(f"Submatrix {k}x{k}: \n{submatrix}\nDeterminant: {determinant}")
            return False
    
    return True
 
A = Matrix(QQ, [
    [4, 12, -16],
    [12, 37, -43],
    [-16, -43, 98]
])
 
is_positive_definite()