P r o j e c t s

N o t e s

T h e D a y

_{I n f o r m a t i o n}
_{g i t h u b}

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.

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