為什麼可逆矩陣又叫「非奇異矩陣(non-singular matrix)」?

最近在撿回之前的線性代數知識，在複習可逆矩陣的時候，發現有的書上把可逆矩陣又稱為非奇異矩陣，乍一看名字完全不知所云，仔細一分析，還是不明白。要想弄明白，還是得從英文入手，下面的解釋主要從這裡得來的Why are invertible matrices called non-singular?。

先把原回答搬過來:

If you take an n×n matrix "at random" (you have to make this very precise, but it can be done sensibly), then it will almost certainly be invertible. That is, the generic case is that of an invertible matrix, the special case is that of a matrix that is not invertible. For example, a 1×1 matrix (with real coefficients) is invertible if and only if it is not the 0 matrix; for 2×2 matrices, it is invertible if and only if the two rows do not lie in the same line through the origin; for 3×3, if and only if the three rows do not lie in the same plane through the origin; etc. So here, "singular" is not being taken in the sense of "single", but rather in the sense of "special", "not common". See the dictionary definition: it includes "odd", "exceptional", "unusual", "peculiar". The noninvertible case is the "special", "uncommon" case for matrices. It is also "singular" in the sense of being the "troublesome" case (you probably know by now that when you are working with matrices, the invertible case is usually the easy one).

主要說了個什麼事呢，意思就是假設隨機生成一個的矩陣，絕大多數情況這個矩陣都是可逆的，也可以理解為它的行列式不為0。換句話說，不可逆的情況是少見的，所以不可逆矩陣就稱為Singular matrix，這裡的singular就是special, not common的意思啊。同理，可逆矩陣很常見，所以就是非奇異矩陣了。