Hermitian symmetry and positive definiteness separately 1 <p>Give any real symmetric or complex Hermitian positive definite \(2\times2\) matrix. Use integers or exact fractions for real and imaginary parts; enter %i for \(i\).</p><p>Half the marks are for Hermitian symmetry and half for positivity of the quadratic form.</p><p>[[input:ansm]] [[validation:ansm]]</p><p>Hermitian symmetry: [[feedback:symmetry]]</p><p>Positivity: [[feedback:positivity]]</p> <p>The checker uses Sylvester’s criterion to test positivity with exact arithmetic. Full credit requires both Hermitian symmetry and positivity. For real entries, Hermitian symmetry means ordinary symmetry.</p> 1 0.1 ansm matrix matrix([1,0],[0,1]) 1 0 1 1 1 1 symmetry 0.5 1 <p>The matrix is Hermitian.</p> 0 PythonChecker ansm true {"checker": "is_hermitian"} = 1 -1 symmetry-T = 0 -1 symmetry-F positivity 0.5 1 <p>The matrix is positive definite.</p> 0 PythonChecker ansm true {"checker": "is_positive_definite"} = 1 -1 positivity-T = 0 -1 positivity-F 1 ansm matrix([1,0],[0,1]) symmetry 1 0 symmetry-T positivity 1 0 positivity-T 2 ansm matrix([2,1],[1,2]) symmetry 1 0 symmetry-T positivity 1 0 positivity-T 3 ansm matrix([1/2,1/4],[1/4,1]) symmetry 1 0 symmetry-T positivity 1 0 positivity-T 4 ansm matrix([-1,0],[0,-1]) symmetry 1 0 symmetry-T positivity 0 0.1 positivity-F 5 ansm matrix([1,1],[1,1]) symmetry 1 0 symmetry-T positivity 0 0.1 positivity-F 6 ansm matrix([1,2],[0,1]) symmetry 0 0.1 symmetry-F positivity 0 0.1 positivity-F 7 ansm matrix([1,2],[2,1]) symmetry 1 0 symmetry-T positivity 0 0.1 positivity-F 8 ansm matrix([2,1],[0,2]) symmetry 0 0.1 symmetry-F positivity 1 0 positivity-T 9 ansm matrix([0,1],[0,0]) symmetry 0 0.1 symmetry-F positivity 0 0.1 positivity-F 10 ansm matrix([2,%i],[-%i,2]) symmetry 1 0 symmetry-T positivity 1 0 positivity-T 11 ansm matrix([1,%i],[%i,1]) symmetry 0 0.1 symmetry-F positivity 1 0 positivity-T 12 ansm matrix([1,2*%i],[-2*%i,1]) symmetry 1 0 symmetry-T positivity 0 0.1 positivity-F 13 ansm matrix([0,%i],[0,0]) symmetry 0 0.1 symmetry-F positivity 0 0.1 positivity-F 14 ansm matrix([1/2,%i/4],[-%i/4,1]) symmetry 1 0 symmetry-T positivity 1 0 positivity-T