What is the geometric meaning of singular matrix

Could anyone help explain what is the geometric meaning of singular matrix ? What's the difference between singular and non-singular matrix ? I know the definition, but couldn't understand it very well.

Thanks.

Solution 1:

If you are in $\Bbb{R}^3$, say you have a matrix like

$$\left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33} \end{array}\right].$$

Now you can think of the columns of this matrix to be the "vectors" corresponding to the sides of a parallelepiped. If this matrix is singular i.e. has determinant zero, then this corresponds to the parallelepiped being completely squashed, a line or just a point.

Solution 2:

A matrix can be thought of as a linear function from a vector space $V$ to a vector space $W$. Typically, one is concerned with $n\times n$ real matrices, which are linear functions from $\mathbb R^n$ to $\mathbb R^n$. An $n\times n$ real matrix is non-singular if its image as a function is all of $\mathbb R^n$ and singular otherwise. More intuitively, it is singular if it misses some point in $n$-dimensional space and non-singular if it doesn't.