Singularity

For an n×nn \times n matrix A=(a1,,an)A = (a_1, \cdots, a_n), the following statements are equivalent.

  • For any vector yy, there is only one vector xx such that y=Axy = Ax.
  • AA is invertible.
  • ImAIm A is not suppressed, which means AA is injective.
  • dimkerA=0\dim \ker A = 0, which means kerA={O}\ker A = \{ O \}.
  • a1,,ana_1, \cdots, a_n are linearly independent.
  • ImAIm A covers the objective space all, which means AA is surjective.
  • rankA=dimImA=nrank A = \dim Im A = n.
  • AA has not the eigenvalues which is zero.
  • So does AtA^t.

The following statements are equivalent.

  • There exists yy such that yAxy \not = Ax for all xx.
  • AA is not invertible.
  • ImAIm A is suppressed, which means AA is not injective.
  • dimkerA>0\dim \ker A > 0, which means kerA\ker A has an element other than OO.
  • a1,,ana_1, \cdots, a_n are linearly dependent.
  • ImAIm A does not cover the objective space all, which means AA is not surjective.
  • rankA=dimImA<nrank A = \dim Im A < n.
  • detA=0\det A = 0.
  • AA has the eigenvalues which is zero.
  • So does AtA^t.

Reference

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


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