What Does the Term Singular Mean in Relation to Square Matrices?

[es]

Why the name "singular"?

The textbook I am currently reading gives the following definition:
"A square matrix is nonsingular if it is the matrix of coefficients of a homogeneous system with a unique solution. It is singular otherwise."

As I understand it a homogeneous system will either have one or
infinitely many solutions and I assumed this is where the choice of
the word singular came from. But if this is the case then the definition
seems backwards. Because the use of the word singular is consistent
in the book I think the definition is not backwards, just my understanding
of the word singular is.

So I went online to try to figure out what singular is describing.
Unfortunately the vast majority of the hits where related to determinates
and matrix inverses. Topics which I have not yet covered.

From the reading I did do I believe an inverse is roughly, a way to get back
to where you came from. So a matrix that can be applied but not unapplied is also singular. This seems like it is probably the significant fact but I am missing the connection to the word singular.

Could someone please explain what the word "singular" is describing?

 

[matt grime]

singular means not invertible. it has nothing to do with the word "single" as in a "single" solution.

 

[LeBrad]

A point where division by zero occurs is called a singularity, and the value at that point is typically not nice. If you're solving the linear system Ax = b to get x = A^-1*b, then if A is singular (non-invertible), you cannot solve it, because you cannot "divide" by A. A is not actually zero and you're not actually dividing by A, but that's how I like to think of singular to help it make sense.

 

[es]

I see now. I was basing my ideas and net searches on the word singular and I should have been using singularity. After making the switch I quickly found this:
http://en.wikipedia.org/wiki/Mathematical_singularity
and coupled with the information provided by Matt and LeBrad I think I understand.

Because there are many solutions to the homogenous system the matrix is not invertible which creates a mathematical singularity. Hence the word singular.

Thanks for the help guys!

 

[mathwonk]

merriam webster says:

singular:

3 : being out of the ordinary : UNUSUAL <on the way home we had a singular adventure>
4 : departing from general usage or expectation : PECULIAR, ODD <the air had a singular chill>
5 a of a matrix : having a determinant equal to zero b of a linear transformation : having the property that the matrix of coefficients of the new variables has a determninant equal to zero.


i.e. it just means something special. i.e. the implication is that the general matrix is invertible.