Ch 2.3 - Characterizations of Invertible Matrices

The Invertible Matrix Theorem (Theorem 8)

• Core Principle: Let $A$ be a square matrix. The following statements are equivalent; that is, for a given $A$, the statements are either all true or all false.

• List of Equivalent Statements (Characterizations of an Invertible Matrix):

  • $A$ is an invertible matrix.

  • $A$ is row equivalent to the identity matrix $(I_n)$. This means $A$ can be reduced to the identity matrix through elementary row operations.

  • $A$ has $n$ pivot positions. This indicates that every column and every row has a pivot.

  • The equation $Ax = 0$ has only the trivial solution ($x = 0$).

  • The columns of $A$ form a linearly independent set.

  • The linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined by $T(x) = Ax$ is one-to-one. This means that if $T(u) = T(v)$, then $u = v$.

  • The equation $Ax = b$ has at least one solution for each $b$ in $\mathbb{R}^n$. This implies that the system is always consistent.

  • The columns of $A$ span $\mathbb{R}^n$. This means that any vector in $\mathbb{R}^n$ can be written as a linear combination of the columns of $A$.

  • The linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ maps $\mathbb{R}^n$ onto $\mathbb{R}^n$. This means the range of $T$ is all of $\mathbb{R}^n$.

  • There is an $n \times n$ matrix $C$ such that $CA = I$. Here, $C$ is the left inverse.

  • There is an $n \times n$ matrix $D$ such that $AD = I$. Here, $D$ is the right inverse.

  • $A^T$ (the transpose of $A$) is an invertible matrix.

Further Implications and Corollaries of The Invertible Matrix Theorem

• Unique Solution for $Ax=b$: Theorem 8 implies that the equation $Ax = b$ has a unique solution for each $b$ in $\mathbb{R}^n$. This statement on its own implies that $A$ is row equivalent to the identity matrix ($I_n$) and, consequently, that $A$ is invertible.

• Product of Invertible Matrices: Let $A$ and $B$ be square matrices. If their product $AB = I$ (the identity matrix), then both $A$ and $B$ are invertible matrices. In this specific case, $B$ is the inverse of $A$ ($B = A^{-1}$), and $A$ is the inverse of $B$ ($A = B^{-1}$). This property is a direct consequence of Theorem 8.

Classification of Square Matrices

• The Invertible Matrix Theorem effectively categorizes the set of all $n \times n$ matrices into two mutually exclusive groups:

  • Invertible (Nonsingular) Matrices: These are matrices that fulfill all the equivalent statements outlined in Theorem 8.

  • Noninvertible (Singular) Matrices: These are matrices that fail to satisfy any (and therefore all) of the equivalent statements in Theorem 8.

Properties of Singular Matrices

• The negation of any statement within the Invertible Matrix Theorem directly describes a characteristic of an $n \times n$ singular matrix.

• Examples of Properties for an $n \times n$ Singular Matrix: Such a matrix:

  • Is not row equivalent to the identity matrix $(I_n)$.

  • Does not have $n$ pivot positions; it will have fewer than $n$ pivots.

  • Has linearly dependent columns.

  • The homogeneous equation $Ax = 0$ has non-trivial (non-zero) solutions.

  • The linear transformation $T(x) = Ax$ is not one-to-one.

  • Its columns do not span $\mathbb{R}^n$.

  • The linear transformation $T(x) = Ax$ does not map $\mathbb{R}^n$ onto $\mathbb{R}^n$. This means there are vectors in the codomain that are not in the image of $T$.

  • The equation $Ax = b$ does not have a unique solution for every $b$ in $\mathbb{R}^n$; it might have infinitely many solutions or no solution for certain $b$.

Scope and Applicability of The Invertible Matrix Theorem

• Exclusively for Square Matrices: It is crucial to remember that the Invertible Matrix Theorem is applicable only to square matrices (matrices where the number of rows equals the number of columns, i.e., $n \times n$ matrices).

• Limitation Example: For instance, if the columns of a non-square matrix, such as a $3 \times 2$ matrix, are linearly independent, we cannot use the Invertible Matrix Theorem to draw conclusions about the existence or non-existence of solutions for equations like $Ax = b$. The properties of the theorem do not universally hold for non-square matrices, even if isolated conditions might appear to be met.

Invertible Linear Transformations

• Relationship with Matrix Multiplication: Matrix multiplication can be directly interpreted as the composition of linear transformations. The equation $A^{-1}AX = X$ visually represents how an inverse transformation "undoes" the original transformation.

• Definition of an Invertible Linear Transformation: A linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is defined as invertible if there exists another 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 $x \in \mathbb{R}^n$

  • The function $S$ is often referred to as the inverse transformation of $T$.

• Theorem 9: Invertibility of Transformation and Standard Matrix: Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a linear transformation, and let $A$ be its standard matrix. Then $T$ is invertible if and only if $A$ is an invertible matrix. This theorem establishes a direct equivalence between the invertibility of a linear transformation and the invertibility of its corresponding standard matrix.

• Inverse Transformation Formula: In the event that $T$ is invertible, the unique inverse linear transformation $S$ is given by the formula $S(x) = A^{-1}x$. This means that applying the inverse matrix $A^{-1}$ to a vector $x$ performs the inverse transformation.