Al though not important for my course, I was curious as to why the condition number of a singular matrix is $\infty$. Which also begs the question of what the condition number of a non-invertible non-square matrix would be.
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The condition number is defined as the ratio between the highest and smallest singular value of the matrix.
If the matrix is singular, then the smallest singular value is $0$.
Thus, the condition numbers tends to $\infty$.
the_candyman
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And what if the matrix is not a square matrix but a a $m \times n$ matrix with no inverse? What would the condition number be? – Mar 14 '21 at 12:54
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1@MariaTakahashi Then, in the definition of the condition number, replace inverse with pseudoinverse. (Condition number of a rectangular matrix) – Vepir Mar 14 '21 at 13:48