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Nonsingular matrices and equivalences

Created over 8 years ago, updated 10 days ago

Nonsingular matrices (also called invertible or nondegenerate matrices) are square matrices that have a multiplicative inverse. They are central to linear algebra because they represent bijective (one-to-one and onto) linear transformations.

An $n \times n$ matrix $A$ is nonsingular if there exists $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$.

The Invertible Matrix Theorem (or Nonsingular Matrix Theorem) provides dozens of equivalent conditions for nonsingularity, connecting concepts from:

  • Linear systems ($Ax = b$ has unique solutions)
  • Vector spaces (columns/rows form a basis)
  • Linear transformations (injectivity, surjectivity, isomorphism)
  • Matrix algebra (existence of inverse, nonzero determinant)
  • Rank and nullity (full rank, trivial null space)
  • Eigenvalues (0 is not an eigenvalue)

The child nodes of this node explore these equivalences in detail.