About the definition of a singular and non-invertible matrix

definitionlinear algebra

In Wolfram Mathworld, they defined a singular matrix as:

A square matrix that does not have a matrix inverse.

So my questions regard this definition are:

  1. Why is the definition of singular matrices restricted to square matrices? Aren't all rectangular matrices singular and non-invertible since they don't have inverses?
  2. When talking about matrix inverse, does it only refer to two-sided inverses and not one-sided inverses?
  3. Can the terms singular and non-invertible be used interchangeably?
  4. The invertible matrix is defined as having a determinant not equal to zero. Do rectangular matrices also have determinants and can this rule be applied to rectangular matrices as well?

Best Answer

  1. You are correct that all non-square matrices are non-invertible. This is why the term "singular" is reserved for the square case: the colloquial meaning of "singular" is "unusual" and non-invertibility is unusual for square matrices but not for non-square matrices. Similarly, we want to be able to say things like "a (square) matrix is singular iff its determinant is nonzero" and we need the matrix to be square to talk about determinants.

  2. "Inverse" always means "two-sided inverse." It happens that a square matrix has a left inverse iff it has a right inverse iff it has an inverse but this isn't true in more general contexts.

  3. Yes, keeping in mind the restriction to square matrices.

  4. An invertible matrix is a matrix with an inverse. It's a theorem that this is equivalent to having a nonzero determinant, for a square matrix. Non-square matrices don't have determinants so this theorem does not apply to them. (However, they do have minors, and you can state a generalization of "invertible iff determinant nonzero" which describes the rank of an arbitrary matrix in terms of the vanishing and nonvanishing of its minors.)

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