nonsingular 非奇异 matrix | invertible| nondegenerate| regular matrix

$$ AB=BA=I_n $$

• There exists the (multiplicative) inverse of $A$:{matrix inversion} $B=A^{-1}$. $B$ is uniquely determined by $A$

Properties of nonsingular matrix

the invertible matrix theorem

Let $A\in K^{n\times n}$, the following statements are equivalent.

• $A$ is inverible. $AB=I=BA$
  • The matrix has inverse under matrix multiplication.$AC=I$
  • $AB=I$
• The linear transformation mapping $f:x\to Ax$ is invertible
  • The linear transformation mapping $f:x\to Ax$ has inverse under function composition.$f\circ g=1=g\circ f$
• $A$ is row-equivalent to the n-by-n identity matrix $I_n$ {they have the same row sapce}
  • column-equivalent
• $rank (A)=n$, full rank
• trivial kernel ,$\ker(A)=\{0\}$
• The linear transformation mapping $x \to Ax$
  • surjective 满射
  • injective 单射
  • bijective 双射
• column →linearly independent
• span (row)=K^{n}
• row , basis→ K^{n}
• $det A \ne 0$
• $A-\lambda I=singular,(A-\lambda I)x=x$, and $\lambda \ne 0$
• the transpose $A^T$ is an invertible matrix

Relation to its adjugate

$$ \mathbf{A}^{-1}=\frac1{\det(\mathbf{A})}\operatorname{adj}(\mathbf{A}). $$