爲何可逆矩陣又叫「非奇異矩陣(non-singular matrix)」?

最近在撿回以前的線性代數知識，在複習可逆矩陣的時候，發現有的書上把可逆矩陣又稱爲非奇異矩陣，乍一看名字徹底不知所云，仔細一分析，仍是不明白。要想弄明白，仍是得從英文入手，下面的解釋主要從這裏得來的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\(n×n\)的矩陣，絕大多數狀況這個矩陣都是可逆的，也能夠理解爲它的行列式不爲0。換句話說，不可逆的狀況是少見的，因此不可逆矩陣就稱爲Singular matrix，這裏的singular就是special, not common的意思啊。同理，可逆矩陣很常見，因此就是非奇異矩陣了。

舉個例子就更好明白了，現假設一個\(1×1\)的矩陣，咱們知道只有這個矩陣等於0的時候纔是不可逆的，其他狀況都是可逆的；再看\(2×2\)的矩陣，這個能夠理解成是一個平面上的兩條線，只要當這兩條線位於通過零點的同一條線上，那麼這個矩陣纔是不可逆的，顯然這種狀況是特殊的；\(3×3\)矩陣同理不加贅述。

MARSGGBO♥原創




2018-11-28