Singularity
For a `ntimes n` matrix `A=(a_1, cdots, a_n)`, the following statements are equivalent.
 For any vector `y`, there is only one vector `x` such that `y=Ax`.
 `A` is invertible.
 `text{Im} A` is not suppresed, which means `A` is injective.
 `dim text{ker} A=0`, which means `text{ker} A={O}`.
 `a_1`, `cdots`, `a_n` are linearly independent.
 `text{Im} A` covers the objective space all, which means `A` is surjective.
 `text{rank} A=dim text{Im} A=n`.
 `det A ne 0`.
 `A` has not the eigenvalue which is zero.
 So does `A^t`.
The following statements are equivalent.
 There exists `y` such that `y ne Ax` for all `x`.
 `A` is not invertible.

`text{Im} A` is suppresed, which means `A` is not injective.

`dim text{ker} A>0`, which means `text{ker} A` has a element other than `O`.

`a_1`, `cdots`, `a_n` are linearly dependent.

`text{Im} A` does not cover the objective space all, which means `A` is not surjective.
 `text{rank} A=dim text{Im} A<n`.

`det A = 0`.

`A` has the eigenvalue which is zero.

So does `A^t`.
︎Reference
[1] Hiraoka Kazuyuki, Hori Gen, Programming No Tame No Senkei Daisu, Ohmsha.
[1] Hiraoka Kazuyuki, Hori Gen, Programming No Tame No Senkei Daisu, Ohmsha.
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