2.3 Characterizations of invertible matrices (可逆矩阵的特征)

本文为《Linear algebra and its applications》的读书笔记

目录

• The Invertible Matrix Theorem 可逆矩阵定理
• Invertible Linear Transformations 可逆线性变换

The Invertible Matrix Theorem 可逆矩阵定理

THEOREM 8

当 $A$A 为 $2\times2$2×2 矩阵时，$(e)$(e) 可以较快的判定矩阵是否为可逆矩阵 (或者计算 $ad-bc$ad−bc)

PROOF
上面的几个结论基本在前面都介绍过了，这里只是总结了一下。唯一没有出现过的就是 $(l)$(l)。下面只证明 $(a)\Leftrightarrow (l)$(a)⇔(l)
$\because (A^T)^{-1}=(A^{-1})^T \\\therefore (a) \Rightarrow (l) \\\because (A)^{-1}=(({A^T})^{-1})^T \\\therefore (l) \Rightarrow (a)$∵(AT)−1=(A−1)T∴(a)⇒(l)∵(A)−1=((AT)−1)T∴(l)⇒(a)

The next fact follows from Theorem 8

由定理 $8$8 的 $(j)(k)$(j)(k) 可以得到上述推论

The Invertible Matrix Theorem divides the set of all $n \times n$n×n matrices into two disjoint (不相交的) classes: the invertible (nonsingular) matrices, and the noninvertible (singular) matrices.

Each statement in the theorem describes a property of every $n \times n$n×n invertible matrix. The negation of a statement (否命题) in the theorem describes a property of every $n \times n$n×n singular matrix. For instance, an $n \times n$n×n singular matrix is not row equivalent to $I_n$In​, does not have $n$n pivot positions, and has linearly dependent columns.

The Invertible Matrix Theorem applies only to square matrices.

EXAMPLE 1
$A$A and $B$B are both $n\times n$n×n martices. Show that if $AB$AB is invertible, so is $A$A.
SOLUTION
Let $W$W be the inverse of $AB$AB. Then $ABW=I$ABW=I, Thus $A$A is invertible.

Invertible Linear Transformations 可逆线性变换

A linear transformation $T : \mathbb R^n \rightarrow \mathbb R^n$T:Rn→Rn is said to be invertible if there exists a function $S : \mathbb R^n \rightarrow \mathbb R^n$S:Rn→Rn such that

The next theorem shows that if such an $S$S exists, it is unique and must be a linear transformation. We call $S$S the inverse of $T$T and write it as $T^{-1}$T−1.

ill-conditiones matrix：病态矩阵
condition number：条件数。条件数大约给出方程 $A\boldsymbol x=\boldsymbol b$Ax=b 的解随 $\boldsymbol b$b 改变的敏感性的粗略度量

本节练习题的 41~45 涉及条件数等概念，可以等学完 6、7 章之后再回过来看