Section 2.3 Characterizations of Invertible Matrices

The Fundamental Role of Pivot Positions in Matrix Algebra

• The properties of an even-sized or rectangular matrix $A$ of size $m \times n$ are fundamentally determined by the locations of its pivot positions.

• Column-Based Pivot Properties:

  • The following statements are true if and only if there is a pivot in every column of $A$:
  • The homogeneous equation $Ax = 0$ has only the trivial solution.
  • The columns of $A$ form a linearly independent set.
  • The linear transformation $x \mapsto Ax$ is one-to-one.

• Row-Based Pivot Properties:

  • The following statements are true if and only if there is a pivot in every row of $A$:
  • The equation $Ax = b$ has a solution for every $b \in \mathbb{R}^m$.
  • The columns of $A$ span $\mathbb{R}^m$.
  • Fresh columns are produced by the transformation $x \mapsto Ax$ mapping $\mathbb{R}^n$ onto $\mathbb{R}^m$.

• The Case for Square Matrices ($m = n$):

  • When the matrix is square, the row and column pivot conditions become interchangeable.
  • A square matrix has a pivot in every column if and only if it has a pivot in every row.

The Invertible Matrix Theorem

• Let $A$ be a square $n \times n$ matrix. The Invertible Matrix Theorem states that the following statements are equivalent (if one is true, all are true; if one is false, all are false):

  • (a) $A$ is an invertible matrix.
  • (b) $A$ is row equivalent to the $n \times n$ identity matrix $I_n$.
  • (c) $A$ has $n$ pivot positions.
  • (d) The equation $Ax = 0$ has only the trivial solution.
  • (e) The columns of $A$ form a linearly independent set.
  • (f) The linear transformation $x \mapsto Ax$ is one-to-one.
  • (g) The equation $Ax = b$ has at least one solution for each $b \in \mathbb{R}^n$.
  • (h) The columns of $A$ span $\mathbb{R}^n$.
  • (i) The linear transformation $x \mapsto Ax$ maps $\mathbb{R}^n$ onto $\mathbb{R}^n$.
  • (j) There is an $n \times n$ matrix $C$ such that $CA = I$.
  • (k) There is an $n \times n$ matrix $D$ such that $AD = I$.
  • (l) $A^T$ is an invertible matrix.

• Implications and Relationships within the Theorem:

  • The theorem establishes a "circle" of implications where each point logically leads to the next.
  • Matrix $D$ in statement (k) works as the inverse, linking (k) and (g) to the main logical circle.
  • Statements (g), (h), and (i) are naturally equivalent for any matrix and are linked through (g) to the core of the theorem.
  • Statements (d), (e), and (f) are equivalent for any matrix $A$ and are linked to the circle through (d).

Categorization of Matrices: Invertible vs. Singular

• The Invertible Matrix Theorem effectively divides the set of all square matrices into two disjoint classes:

  • Invertible (Nonsingular) Matrices: These possess every property listed in the theorem.
  • Noninvertible (Singular) Matrices: These possess none of the properties listed. For example, an $n \times n$ singular matrix is not row equivalent to $I_n$, does not have $n$ pivot positions, and has linearly dependent columns.

• Extension to Transposes:

  • If $A$ is invertible, then $A^T$ is also invertible and $(A^T)^{-1} = (A^{-1})^T$.

Practical Application and Limitations

• Example 1: Assessing Invertibility via Pivots

  • Consider the matrix:     $A = \begin{pmatrix} 1 & 0 & -2 \\ 3 & 1 & -2 \\ -5 & -1 & 9 \end{pmatrix}$
  • By performing row operations, we determine if $A$ has three pivot positions.
  • Step 1: Add $-3$ times row 1 to row 2, and add $5$ times row 1 to row 3:     $\begin{pmatrix} 1 & 0 & -2 \\ 0 & 1 & 4 \\ 0 & -1 & -1 \end{pmatrix}$
  • Step 2: Add row 2 to row 3:     $\begin{pmatrix} 1 & 0 & -2 \\ 0 & 1 & 4 \\ 0 & 0 & 3 \end{pmatrix}$
  • Since the result is in echelon form and shows three pivots, matrix $A$ is invertible according to statement (c) of the Invertible Matrix Theorem.

• Critical Restriction: Square Matrices Only

  • The Invertible Matrix Theorem applies strictly to square matrices.
  • If a rectangular matrix (e.g., $4 \times 3$) has linearly independent columns, one cannot use the Invertible Matrix Theorem to conclude anything about the existence of solutions for $Ax = b$.

Invertible Linear Transformations

• Conceptual Mapping:

  • Matrix multiplication corresponds to the composition of linear transformations.
  • If matrix $A$ is invertible, then for the equation $y = Ax$, the operation can be reversed: $x = A^{-1}y$. This means $A^{-1}$ transforms $Ax$ back into $x$.

• Definition of an Invertible Transformation:

  • A linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is invertible if there exists a function $S: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that:
  1. $S(T(x)) = x$ for all $x \in \mathbb{R}^n$
  2. $T(S(x)) = x$ for all $x \in \mathbb{R}^n$

• Theorem 9:

  • Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a linear transformation and let $A$ be the standard matrix for $T$.
  • $T$ is invertible if and only if $A$ is an invertible matrix.
  • In this scenario, the unique inverse function $S$ is defined by $S(x) = A^{-1}x$.

• Proof Overview for Theorem 9:

  • If $T$ is invertible, then for any $b \in \mathbb{R}^n$, we have $b = T(S(b))$, indicating that every $b$ is in the range of $T$. Thus, $T$ is onto. By statement (i) of the Invertible Matrix Theorem, the matrix $A$ must be invertible.
  • Conversely, if $A$ is invertible, defining $S(x) = A^{-1}x$ provides a linear transformation that satisfies the requirements for an inverse, such as $S(T(x)) = S(Ax) = A^{-1}(Ax) = x$.