# P r o j e c t s

N o t e s

## 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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