We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as "matrix multiplication." This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

## Matrix-Vector Product

We have repeatedly seen the importance of forming linear combinations of the columns of a matrix. As one example of this, the oft-used Theorem SLSLC, said that every solution to a system of linear equations gives rise to a linear combination of the column vectors of the coefficient matrix that equals the vector of constants. This theorem, and others, motivate the following central definition.

Definition MVP (Matrix-Vector Product) Suppose $A$ is an $m\times n$ matrix with columns $\vectorlist{A}{n}$ and $\vect{u}$ is a vector of size $n$. Then the matrix-vector product of $A$ with $\vect{u}$ is the linear combination \begin{equation*} A\vect{u}= \vectorentry{\vect{u}}{1}\vect{A}_1+ \vectorentry{\vect{u}}{2}\vect{A}_2+ \vectorentry{\vect{u}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{u}}{n}\vect{A}_n \end{equation*}

So, the matrix-vector product is yet another version of "multiplication," at least in the sense that we have yet again overloaded juxtaposition of two symbols as our notation. Remember your objects, an $m\times n$ matrix times a vector of size $n$ will create a vector of size $m$. So if $A$ is rectangular, then the size of the vector changes. With all the linear combinations we have performed so far, this computation should now seem second nature.

Example MTV: A matrix times a vector.

We can now represent systems of linear equations compactly with a matrix-vector product (Definition MVP) and column vector equality (Definition CVE). This finally yields a very popular alternative to our unconventional $\linearsystem{A}{\vect{b}}$ notation.

Theorem SLEMM (Systems of Linear Equations as Matrix Multiplication) The set of solutions to the linear system $\linearsystem{A}{\vect{b}}$ equals the set of solutions for $\vect{x}$ in the vector equation $A\vect{x}=\vect{b}$.

Example MNSLE: Matrix notation for systems of linear equations.

The matrix-vector product is a very natural computation. We have motivated it by its connections with systems of equations, but here is a another example.

Example MBC: Money's best cities.

Later (much later) we will need the following theorem, which is really a technical lemma (see technique LC). Since we are in a position to prove it now, we will. But you can safely skip it for the moment, if you promise to come back later to study the proof when the theorem is employed. At that point you will also be able to understand the comments in the paragraph following the proof.

Theorem EMMVP (Equal Matrices and Matrix-Vector Products) Suppose that $A$ and $B$ are $m\times n$ matrices such that $A\vect{x}=B\vect{x}$ for every $\vect{x}\in\complex{n}$. Then $A=B$.

You might notice that the hypotheses of this theorem could be "weakened" (i.e.  made less restrictive). We could suppose the equality of the matrix-vector products for just the standard unit vectors (Definition SUV) or any other spanning set (Definition TSVS) of $\complex{n}$ (exercise LISS.T40). However, in practice, when we apply this theorem we will only need this weaker form. (If we made the hypothesis less restrictive, we would call the theorem "stronger.")

## Matrix Multiplication

We now define how to multiply two matrices together. Stop for a minute and think about how you might define this new operation.

Many books would present this definition much earlier in the course. However, we have taken great care to delay it as long as possible and to present as many ideas as practical based mostly on the notion of linear combinations. Towards the conclusion of the course, or when you perhaps take a second course in linear algebra, you may be in a position to appreciate the reasons for this. For now, understand that matrix multiplication is a central definition and perhaps you will appreciate its importance more by having saved it for later.

Definition MM (Matrix Multiplication) Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix with columns $\vectorlist{B}{p}$. Then the matrix product of $A$ with $B$ is the $m\times p$ matrix where column $i$ is the matrix-vector product $A\vect{B}_i$. Symbolically, \begin{equation*} AB=A\matrixcolumns{B}{p}=\left[A\vect{B}_1|A\vect{B}_2|A\vect{B}_3|\ldots|A\vect{B}_p\right]. \end{equation*}

Example PTM: Product of two matrices.

Is this the definition of matrix multiplication you expected? Perhaps our previous operations for matrices caused you to think that we might multiply two matrices of the same size, {\em entry-by-entry}? Notice that our current definition uses matrices of different sizes (though the number of columns in the first must equal the number of rows in the second), and the result is of a third size. Notice too in the previous example that we cannot even consider the product $BA$, since the sizes of the two matrices in this order aren't right.

But it gets weirder than that. Many of your old ideas about "multiplication" won't apply to matrix multiplication, but some still will. So make no assumptions, and don't do anything until you have a theorem that says you can. Even if the sizes are right, matrix multiplication is not commutative --- order matters.

Example MMNC: Matrix multiplication is not commutative.

## Matrix Multiplication, Entry-by-Entry

While certain "natural" properties of multiplication don't hold, many more do. In the next subsection, we'll state and prove the relevant theorems. But first, we need a theorem that provides an alternate means of multiplying two matrices. In many texts, this would be given as the definition of matrix multiplication. We prefer to turn it around and have the following formula as a consequence of our definition. It will prove useful for proofs of matrix equality, where we need to examine products of matrices, entry-by-entry.

Theorem EMP (Entries of Matrix Products) Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Then for $1\leq i\leq m$, $1\leq j\leq p$, the individual entries of $AB$ are given by

\begin{align*} \matrixentry{AB}{ij} &= \matrixentry{A}{i1}\matrixentry{B}{1j}+ \matrixentry{A}{i2}\matrixentry{B}{2j}+ \matrixentry{A}{i3}\matrixentry{B}{3j}+ \cdots+ \matrixentry{A}{in}\matrixentry{B}{nj}\\ &= \sum_{k=1}^{n}\matrixentry{A}{ik}\matrixentry{B}{kj} \end{align*}

Example PTMEE: Product of two matrices, entry-by-entry.

Theorem EMP is the way many people compute matrix products by hand. It will also be very useful for the theorems we are going to prove shortly. However, the definition (Definition MM) is frequently the most useful for its connections with deeper ideas like the null space and the upcoming column space.

## Properties of Matrix Multiplication

In this subsection, we collect properties of matrix multiplication and its interaction with the zero matrix (Definition ZM), the identity matrix (Definition IM), matrix addition (Definition MA), scalar matrix multiplication (Definition MSM), the inner product (Definition IP), conjugation (Theorem MMCC), and the transpose (Definition TM). Whew! Here we go. These are great proofs to practice with, so try to concoct the proofs before reading them, they'll get progressively more complicated as we go.

Theorem MMZM (Matrix Multiplication and the Zero Matrix) Suppose $A$ is an $m\times n$ matrix. Then

1. $A\zeromatrix_{n\times p}=\zeromatrix_{m\times p}$
2. $\zeromatrix_{p\times m}A=\zeromatrix_{p\times n}$

Theorem MMIM (Matrix Multiplication and Identity Matrix) Suppose $A$ is an $m\times n$ matrix. Then

1. $AI_n=A$
2. $I_mA=A$

It is this theorem that gives the identity matrix its name. It is a matrix that behaves with matrix multiplication like the scalar 1 does with scalar multiplication. To multiply by the identity matrix is to have no effect on the other matrix.

Theorem MMDAA (Matrix Multiplication Distributes Across Addition) Suppose $A$ is an $m\times n$ matrix and $B$ and $C$ are $n\times p$ matrices and $D$ is a $p\times s$ matrix. Then

1. $A(B+C)=AB+AC$
2. $(B+C)D=BD+CD$

Theorem MMSMM (Matrix Multiplication and Scalar Matrix Multiplication) Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Let $\alpha$ be a scalar. Then $\alpha(AB)=(\alpha A)B=A(\alpha B)$.

Theorem MMA (Matrix Multiplication is Associative ) Suppose $A$ is an $m\times n$ matrix, $B$ is an $n\times p$ matrix and $D$ is a $p\times s$ matrix. Then $A(BD)=(AB)D$.

The statement of our next theorem is technically inaccurate. If we upgrade the vectors $\vect{u},\,\vect{v}$ to matrices with a single column, then the expression $\transpose{\vect{u}}\conjugate{\vect{v}}$ is a $1\times 1$ matrix, though we will treat this small matrix as if it was simply the scalar quantity in its lone entry. When we apply Theorem MMIP there should not be any confusion.

Theorem MMIP (Matrix Multiplication and Inner Products) If we consider the vectors $\vect{u},\,\vect{v}\in\complex{m}$ as $m\times 1$ matrices then \begin{equation*} \innerproduct{\vect{u}}{\vect{v}}=\transpose{\vect{u}}\conjugate{\vect{v}} \end{equation*}

Theorem MMCC (Matrix Multiplication and Complex Conjugation) Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Then $\conjugate{AB}=\conjugate{A}\,\conjugate{B}$.

Another theorem in this style, and its a good one. If you've been practicing with the previous proofs you should be able to do this one yourself.

Theorem MMT (Matrix Multiplication and Transposes) Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Then $\transpose{(AB)}=\transpose{B}\transpose{A}$.

This theorem seems odd at first glance, since we have to switch the order of $A$ and $B$. But if we simply consider the sizes of the matrices involved, we can see that the switch is necessary for this reason alone. That the individual entries of the products then come along to be equal is a bonus.

As the adjoint of a matrix is a composition of a conjugate and a transpose, its interaction with matrix multiplication is similar to that of a transpose. Here's the last of our long list of basic properties of matrix multiplication.

Theorem MMAD (Matrix Multiplication and Adjoints) Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Then $\adjoint{(AB)}=\adjoint{B}\adjoint{A}$.

Notice how none of these proofs above relied on writing out huge general matrices with lots of ellipses ("...") and trying to formulate the equalities a whole matrix at a time. This messy business is a "proof technique" to be avoided at all costs. Notice too how the proof of Theorem MMAD does not use an entry-by-entry approach, but simply builds on previous results about matrix multiplication's interaction with conjugation and transposes.

These theorems, along with Theorem VSPM and the other results in Section MO:Matrix Operations, give you the "rules" for how matrices interact with the various operations we have defined on matrices (addition, scalar multiplication, matrix multiplication, conjugation, transposes and adjoints). Use them and use them often. But don't try to do anything with a matrix that you don't have a rule for. Together, we would informally call all these operations, and the attendant theorems, "the algebra of matrices." Notice, too, that every column vector is just a $n\times 1$ matrix, so these theorems apply to column vectors also. Finally, these results, taken as a whole, may make us feel that the definition of matrix multiplication is not so unnatural.

## Hermitian Matrices

The adjoint of a matrix has a basic property when employed in a matrix-vector product as part of an inner product. At this point, you could even use the following result as a motivation for the definition of an adjoint.

Theorem AIP (Adjoint and Inner Product) Suppose that $A$ is an $m\times n$ matrix and $\vect{x}\in\complex{n}$, $\vect{y}\in\complex{m}$. Then $\innerproduct{A\vect{x}}{\vect{y}}=\innerproduct{\vect{x}}{\adjoint{A}\vect{y}}$.

Sometimes a matrix is equal to its adjoint (Definition A), and these matrices have interesting properties. One of the most common situations where this occurs is when a matrix has only real number entries. Then we are simply talking about symmetric matrices (Definition SYM), so you can view this as a generalization of a symmetric matrix.

Definition HM (Hermitian Matrix) The square matrix $A$ is Hermitian (or self-adjoint) if $A=\adjoint{A}$.

Again, the set of real matrices that are Hermitian is exactly the set of symmetric matrices. In Section PEE:Properties of Eigenvalues and Eigenvectors we will uncover some amazing properties of Hermitian matrices, so when you get there, run back here to remind yourself of this definition. Further properties will also appear in various sections of the Topics (part T). Right now we prove a fundamental result about Hermitian matrices, matrix vector products and inner products. As a characterization, this could be employed as a definition of a Hermitian matrix and some authors take this approach.

Theorem HMIP (Hermitian Matrices and Inner Products) Suppose that $A$ is a square matrix of size $n$. Then $A$ is Hermitian if and only if $\innerproduct{A\vect{x}}{\vect{y}}=\innerproduct{\vect{x}}{A\vect{y}}$ for all $\vect{x},\,\vect{y}\in\complex{n}$.

So, informally, Hermitian matrices are those that can be tossed around from one side of an inner product to the other with reckless abandon. We'll see later what this buys us.