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  • 非奇异的;非退化的;满秩

     小结:

    1、非奇异的;非退化的:只有输入为0(可以泛化的概念)结果才为0

     2、If A ∈ Mm,n(F) and m < n, then A is
    necessarily singular. 

    A s*n s<n 则A必然奇异的。

    A linear transformation or matrix is said to be nonsingular if it produces the output 0 only for the input 0.

    Matrix Analysis p14

    A linear transformation or matrix is said to be nonsingular if it produces the output 0
    only for the input 0. Otherwise, it is singular. If A ∈ Mm,n(F) and m < n, then A is
    necessarily singular. An A ∈ Mn(F) is invertible if there is a matrix A−1 ∈ Mn(F) (the
    inverse of A) such that A−1A = I. If A ∈ Mn and A−1A = I, then AA−1 = I; that is,
    A−1 is a left inverse if and only if it is a right inverse; A−1 is unique whenever it exists.
    It is useful to be able to call on a variety of criteria for a square matrix to be
    nonsingular. The following are equivalent for a given A ∈ Mn(F):
    (a) A is nonsingular.
    (b) A−1 exists.
    (c) rank A = n.
    (d) The rows of A are linearly independent.
    (e) The columns of A are linearly independent.
    (f) det A /= 0.
    (g) The dimension of the range of A is n.
    (h) The dimension of the null space of A is 0.
    (i) Ax = b is consistent for each b ∈ Fn.
    (j) If Ax = b is consistent, then the solution is unique.
    (k) Ax = b has a unique solution for each b ∈ Fn.
    (l) The only solution to Ax = 0 is x = 0.
    (m) 0 is not an eigenvalue of A (see Chapter 1).

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  • 原文地址:https://www.cnblogs.com/rsapaper/p/10557062.html
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