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.

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You can think of a n×n

normal matrix as a linear transformation. It ‘stretches’ or ‘scale’ a vector in each of its eigenvector directions by factor of eigenvalues, which could be a complex number.

If there are m

zero eigenvalues and the m independent corresponding eigenvectors, you can interpret it as the matrix wipes out m dimensions in the n dimensional vector space, or m dimensions are squashed or collapse, as described in BenjaLim’s answer. The annihilated m dimensions is called the null space or kernel of the matrix, and the remained n−m dimensions is the co-image of the matrix, whose dimension (n−m

in this case) is the rank of the matrix.

If the matrix has null space (zero eigenvalues), you cannot invert it because information in the null space are lost, just as you cannot invert 0×a=b

by a=b/0. The best you can do is to recover the information in the co-image space by pseudo-inverse.