Singularity

For an $n \times n$ 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.
• $Im A$ is not suppressed, which means $A$ is injective.
• $\dim \ker A = 0$, which means $\ker A = \{ O \}$.
• $a_1, \cdots, a_n$ are linearly independent.
• $Im A$ covers the objective space all, which means $A$ is surjective.
• $rank A = \dim Im A = n$.
• $A$ has not the eigenvalues which is zero.
• So does $A^t$.

The following statements are equivalent.

• There exists $y$ such that $y \not = Ax$ for all $x$.
• $A$ is not invertible.
• $Im A$ is suppressed, which means $A$ is not injective.
• $\dim \ker A > 0$, which means $\ker A$ has an element other than $O$.
• $a_1, \cdots, a_n$ are linearly dependent.
• $Im A$ does not cover the objective space all, which means $A$ is not surjective.
• $rank A = \dim Im A < n$.
• $\det A = 0$.
• $A$ has the eigenvalues which is zero.
• So does $A^t$.

Reference

[1] Hiraoka Kazuyuki, Hori Gen, Programming No Tame No Senkei Daisu, Ohmsha.