二維逆問題/單調性

完全正性

A totally positive matrix is a matrix in which the determinant of every square submatrix, is positive. Certain submatrices of kernels and matrices of Dirichlet-to-Neumann operators are totally positive. In fact the total positivity essentially characterizes the matrices and that allows one to obtain the valid data for discrete inverse problems from the continuous one, see [CIM], [CMM] and [IM]. 

練習 (***). 使用高斯-約旦消元法證明每個平方完全正矩陣可以分解成以下簡單型別的矩陣

${\begin{pmatrix}1&0&\ldots &0\\0&x&\ldots &0\\\vdots &\vdots &\ddots &\ldots \\0&0&\ldots &1\end{pmatrix}}{\mbox{ and}}{\begin{pmatrix}1&0&\ldots &0\\x&1&\ldots &0\\\vdots &\vdots &\ddots &\ldots \\0&0&\ldots &1\end{pmatrix}},$

其中 x > 0.

練習 (**). 使用上一個練習來證明，用完全正矩陣乘以向量會減少向量符號變化的次數。也就是說，完全正矩陣具有單調性。

One can prove the total positivity property of restrictions of kernels of planar domains using the variation diminishing property or by approximation by planar networks. The rotation invariance and the total positivity of the kernels together are equivalent to the convolutions functions being positive-definite and are completely characterized by Bochner theorem.

複合矩陣

The compound matrix is an important construction for the study of totally positive matrices, see [GK]. For a given matrix M the compound matrix C of order n is the matrix which entries are equal to the determinants of the n by n square submatrices of M arranged in the lexicographical order. Therefore, a matrix M is totally positive if and only if its compound matrices of all orders have positive entries.

由柯西-比內公式得出

$C(MN)=C(M)(C(N)$

由於對角矩陣的複合矩陣也是對角矩陣，因此可以從原始矩陣的譜分解中獲得複合矩陣的譜分解。也就是說，如果

$M=SDS^{-1},$

那麼

$C(M)=C(S)C(D)C(S)^{-1}.$

練習 (*). 設 M 為 n 乘 n 方陣，C_k(M) 為其 k 階複合矩陣。用 C_n 和 C_n-1 的元素表示 M 的逆矩陣 M^-1 的元素。

練習 (**). 假設 M 是一個方陣，具有如上的譜分解。證明其 n 階複合矩陣 C(M) 的特徵值為原始矩陣所有可能的 n 元組特徵值的乘積。

譜特性

The spectrum of a square totally positive matrix is simple. That is, all its eigenvalues are positive and have multiplicity one. 

練習 (***). 將佩龍-弗羅貝尼烏斯定理應用於完全正矩陣的複合矩陣，以證明上面的陳述，參見 [GK]。

完全正矩陣的特徵向量形成線性無關的切比雪夫系統。