In mathematics, the determinantal conjecture of and asks whether the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of a sum ''A'' + ''B'' of two ''n'' by ''n''
normal complex matrices ''A'' and ''B'' lies in the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of the ''n''! points Π
''i'' (''λ(A)''
''i'' + ''λ(B)''
σ(''i'')), where the numbers ''λ(A)''
''i'' and ''λ(B)''
''i'' are the
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of ''A'' and ''B'', and ''σ'' is an element of the symmetric group ''S''
''n''.
References
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Determinants
Conjectures
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