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).