We saw in Theorem CINM that if a square matrix $A$ is nonsingular, then there is a matrix $B$ so that $AB=I_n$. In other words, $B$ is halfway to being an inverse of $A$. We will see in this section that $B$ automatically fulfills the second condition ($BA=I_n$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We'll make all these connections precise now. Not many examples or definitions in this section, just theorems.

## Nonsingular Matrices are Invertible

We need a couple of technical results for starters. Some books would call these minor, but essential, results "lemmas." We'll just call 'em theorems.

The first of these technical results is interesting in that the hypothesis says something about a product of two square matrices and the conclusion then says the same thing about each individual matrix in the product. This result has an analogy in the algebra of complex numbers: suppose $\alpha,\,\beta\in\complexes$, then $\alpha\beta\neq 0$ if and only if $\alpha\neq 0$ and $\beta\neq 0$. We can view this result as suggesting that the term "nonsingular" for matrices is like the term "nonzero" for scalars.

Theorem NPNT (Nonsingular Product has Nonsingular Terms) Suppose that $A$ and $B$ are square matrices of size $n$. The product $AB$ is nonsingular if and only if $A$ and $B$ are both nonsingular.

This is a powerful result in the "forward" direction, because it allows us to begin with a hypothesis that something complicated (the matrix product $AB$) has the property of being nonsingular, and we can then conclude that the simpler constituents ($A$ and $B$ individually) then also have the property of being nonsingular. If we had thought that the matrix product was an artificial construction, results like this would make us begin to think twice.

The contrapositive of this result is equally interesting. It says that $A$ or $B$ (or both) is a singular matrix if and only if the product $AB$ is singular. Notice how the negation of the theorem's conclusion ($A$ and $B$ both nonsingular) becomes the statement "at least one of $A$ and $B$ is singular." (See technique CP.)

Theorem OSIS (One-Sided Inverse is Sufficient) Suppose $A$ and $B$ are square matrices of size $n$ such that $AB=I_n$. Then $BA=I_n$.

So Theorem OSIS tells us that if $A$ is nonsingular, then the matrix $B$ guaranteed by Theorem CINM will be both a "right-inverse" and a "left-inverse" for $A$, so $A$ is invertible and $\inverse{A}=B$.

So if you have a nonsingular matrix, $A$, you can use the procedure described in Theorem CINM to find an inverse for $A$. If $A$ is singular, then the procedure in Theorem CINM will fail as the first $n$ columns of $M$ will not row-reduce to the identity matrix. However, we can say a bit more. When $A$ is singular, then $A$ does not have an inverse (which is very different from saying that the procedure in Theorem CINM fails to find an inverse). This may feel like we are splitting hairs, but its important that we do not make unfounded assumptions. These observations motivate the next theorem.

Theorem NI (Nonsingularity is Invertibility) Suppose that $A$ is a square matrix. Then $A$ is nonsingular if and only if $A$ is invertible.

So for a square matrix, the properties of having an inverse and of having a trivial null space are one and the same. Can't have one without the other.

Theorem NME3 (Nonsingular Matrix Equivalences, Round 3) Suppose that $A$ is a square matrix of size $n$. The following are equivalent.

1. $A$ is nonsingular.
2. $A$ row-reduces to the identity matrix.
3. The null space of $A$ contains only the zero vector, $\nsp{A}=\set{\zerovector}$.
4. The linear system $\linearsystem{A}{\vect{b}}$ has a unique solution for every possible choice of $\vect{b}$.
5. The columns of $A$ are a linearly independent set.
6. $A$ is invertible.

In the case that $A$ is a nonsingular coefficient matrix of a system of equations, the inverse allows us to very quickly compute the unique solution, for any vector of constants.

Theorem SNCM (Solution with Nonsingular Coefficient Matrix) Suppose that $A$ is nonsingular. Then the unique solution to $\linearsystem{A}{\vect{b}}$ is $\inverse{A}\vect{b}$.

## Unitary Matrices

Recall that the adjoint of a matrix is $\adjoint{A}=\transpose{\left(\conjugate{A}\right)}$ (Definition A).

Definition UM (Unitary Matrices) Suppose that $U$ is a square matrix of size $n$ such that $\adjoint{U}U=I_n$. Then we say $U$ is unitary.

This condition may seem rather far-fetched at first glance. Would there be any matrix that behaved this way? Well, yes, here's one.

Example UM3: Unitary matrix of size 3.

Unitary matrices do not have to look quite so gruesome. Here's a larger one that is a bit more pleasing.

Example UPM: Unitary permutation matrix.

If a matrix $A$ has only real number entries (we say it is a real matrix) then the defining property of being unitary simplifies to $\transpose{A}A=I_n$. In this case we, and everybody else, calls the matrix orthogonal, so you may often encounter this term in your other reading when the complex numbers are not under consideration.

Unitary matrices have easily computed inverses. They also have columns that form orthonormal sets. Here are the theorems that show us that unitary matrices are not as strange as they might initially appear.

Theorem UMI (Unitary Matrices are Invertible) Suppose that $U$ is a unitary matrix of size $n$. Then $U$ is nonsingular, and $\inverse{U}=\adjoint{U}$.

Theorem CUMOS (Columns of Unitary Matrices are Orthonormal Sets) Suppose that $A$ is a square matrix of size $n$ with columns $S=\set{\vectorlist{A}{n}}$. Then $A$ is a unitary matrix if and only if $S$ is an orthonormal set.

Example OSMC: Orthonormal set from matrix columns.

When using vectors and matrices that only have real number entries, orthogonal matrices are those matrices with inverses that equal their transpose. Similarly, the inner product is the familiar dot product. Keep this special case in mind as you read the next theorem.

Theorem UMPIP (Unitary Matrices Preserve Inner Products) Suppose that $U$ is a unitary matrix of size $n$ and $\vect{u}$ and $\vect{v}$ are two vectors from $\complex{n}$. Then

\begin{align*} \innerproduct{U\vect{u}}{U\vect{v}}&=\innerproduct{\vect{u}}{\vect{v}} & &\text{and} & \norm{U\vect{v}}&=\norm{\vect{v}} \end{align*}

Aside from the inherent interest in this theorem, it makes a bigger statement about unitary matrices. When we view vectors geometrically as directions or forces, then the norm equates to a notion of length. If we transform a vector by multiplication with a unitary matrix, then the length (norm) of that vector stays the same. If we consider column vectors with two or three slots containing only real numbers, then the inner product of two such vectors is just the dot product, and this quantity can be used to compute the angle between two vectors. When two vectors are multiplied (transformed) by the same unitary matrix, their dot product is unchanged and their individual lengths are unchanged. The results in the angle between the two vectors remaining unchanged.

A "unitary transformation" (matrix-vector products with unitary matrices) thus preserve geometrical relationships among vectors representing directions, forces, or other physical quantities. In the case of a two-slot vector with real entries, this is simply a rotation. These sorts of computations are exceedingly important in computer graphics such as games and real-time simulations, especially when increased realism is achieved by performing many such computations quickly. We will see unitary matrices again in subsequent sections (especially Theorem OD) and in each instance, consider the interpretation of the unitary matrix as a sort of geometry-preserving transformation. Some authors use the term isometry to highlight this behavior. We will speak loosely of a unitary matrix as being a sort of generalized rotation.

A final reminder: the terms "dot product," "symmetric matrix" and "orthogonal matrix" used in reference to vectors or matrices with real number entries correspond to the terms "inner product," "Hermitian matrix" and "unitary matrix" when we generalize to include complex number entries, so keep that in mind as you read elsewhere.