In this section we will back up and start simple. First a definition of a totally general set of matrices.

Definition VSM (Vector Space of $m\times n$ Matrices) The vector space $M_{mn}$ is the set of all $m\times n$ matrices with entries from the set of complex numbers.

Just as we made, and used, a careful definition of equality for column vectors, so too, we have precise definitions for matrices.

Definition ME (Matrix Equality) The $m\times n$ matrices $A$ and $B$ are equal, written $A=B$ provided $\matrixentry{A}{ij}=\matrixentry{B}{ij}$ for all $1\leq i\leq m$, $1\leq j\leq n$.
(This definition contains Notation ME.)

So equality of matrices translates to the equality of complex numbers, on an entry-by-entry basis. Notice that we now have yet another definition that uses the symbol "=" for shorthand. Whenever a theorem has a conclusion saying two matrices are equal (think about your objects), we will consider appealing to this definition as a way of formulating the top-level structure of the proof. We will now define two operations on the set $M_{mn}$. Again, we will overload a symbol (`+') and a convention (juxtaposition for scalar multiplication).

Definition MA (Matrix Addition) Given the $m\times n$ matrices $A$ and $B$, define the sum of $A$ and $B$ as an $m\times n$ matrix, written $A+B$, according to

\begin{align*} \matrixentry{A+B}{ij}&=\matrixentry{A}{ij}+\matrixentry{B}{ij} &&1\leq i\leq m,\,1\leq j\leq n \end{align*}

(This definition contains Notation MA.)

So matrix addition takes two matrices of the same size and combines them (in a natural way!) to create a new matrix of the same size. Perhaps this is the "obvious" thing to do, but it doesn't relieve us from the obligation to state it carefully.

Example MA: Addition of two matrices in $M_{23}$.

Our second operation takes two objects of different types, specifically a number and a matrix, and combines them to create another matrix. As with vectors, in this context we call a number a scalar in order to emphasize that it is not a matrix.

Definition MSM (Matrix Scalar Multiplication) Given the $m\times n$ matrix $A$ and the scalar $\alpha\in\complex{\null}$, the scalar multiple of $A$ is an $m\times n$ matrix, written $\alpha A$ and defined according to

\begin{align*} \matrixentry{\alpha A}{ij}&=\alpha\matrixentry{A}{ij}&&     1\leq i\leq m,\,1\leq j\leq n \end{align*}

(This definition contains Notation MSM.)

Notice again that we have yet another kind of multiplication, and it is again written putting two symbols side-by-side. Computationally, scalar matrix multiplication is very easy.

Example MSM: Scalar multiplication in $M_{32}$.

Vector Space Properties

With definitions of matrix addition and scalar multiplication we can now state, and prove, several properties of each operation, and some properties that involve their interplay. We now collect ten of them here for later reference.

Theorem VSPM (Vector Space Properties of Matrices) Suppose that $M_{mn}$ is the set of all $m\times n$ matrices (Definition VSM) with addition and scalar multiplication as defined in Definition MA and Definition MSM. Then

• If $A,\,B\in M_{mn}$, then $A+B\in M_{mn}$.
• If $\alpha\in\complex{\null}$ and $A\in M_{mn}$, then $\alpha A\in M_{mn}$.
• If $A,\,B\in M_{mn}$, then $A+B=B+A$.
• If $A,\,B,\,C\in M_{mn}$, then $A+\left(B+C\right)=\left(A+B\right)+C$.
• There is a matrix, $\zeromatrix$, called the zero matrix, such that $A+\zeromatrix=A$ for all $A\in M_{mn}$.
• If $A\in M_{mn}$, then there exists a matrix $-A\in M_{mn}$ so that $A+(-A)=\zeromatrix$.
• If $\alpha,\,\beta\in\complex{\null}$ and $A\in M_{mn}$, then $\alpha(\beta A)=(\alpha\beta)A$.
• If $\alpha\in\complex{\null}$ and $A,\,B\in M_{mn}$, then $\alpha(A+B)=\alpha A+\alpha B$.
• If $\alpha,\,\beta\in\complex{\null}$ and $A\in M_{mn}$, then $(\alpha+\beta)A=\alpha A+\beta A$.
• If $A\in M_{mn}$, then $1A=A$.

For now, note the similarities between Theorem VSPM about matrices and Theorem VSPCV about vectors.

The zero matrix described in this theorem, $\zeromatrix$, is what you would expect --- a matrix full of zeros.

Definition ZM (Zero Matrix) The $m\times n$ zero matrix is written as $\zeromatrix=\zeromatrix_{m\times n}$ and defined by $\matrixentry{\zeromatrix}{ij}=0$, for all $1\leq i\leq m$, $1\leq j\leq n$.
(This definition contains Notation ZM.)

Transposes and Symmetric Matrices

We describe one more common operation we can perform on matrices. Informally, to transpose a matrix is to build a new matrix by swapping its rows and columns.

Definition TM (Transpose of a Matrix) Given an $m\times n$ matrix $A$, its transpose is the $n\times m$ matrix $\transpose{A}$ given by \begin{equation*} \matrixentry{\transpose{A}}{ij}=\matrixentry{A}{ji},    1\leq i\leq n,\,1\leq j\leq m. \end{equation*}

Example TM: Transpose of a $3\times 4$ matrix.

It will sometimes happen that a matrix is equal to its transpose. In this case, we will call a matrix symmetric. These matrices occur naturally in certain situations, and also have some nice properties, so it is worth stating the definition carefully. Informally a matrix is symmetric if we can "flip" it about the main diagonal (upper-left corner, running down to the lower-right corner) and have it look unchanged.

Definition SYM (Symmetric Matrix) The matrix $A$ is symmetric if $A=\transpose{A}$.

Example SYM: A symmetric $5\times 5$ matrix.

You might have noticed that Definition SYM did not specify the size of the matrix $A$, as has been our custom. That's because it wasn't necessary. An alternative would have been to state the definition just for square matrices, but this is the substance of the next proof.

Theorem SMS (Symmetric Matrices are Square) Suppose that $A$ is a symmetric matrix. Then $A$ is square.

We finish this section with three easy theorems, but they illustrate the interplay of our three new operations, our new notation, and the techniques used to prove matrix equalities.

Theorem TMA (Transpose and Matrix Addition) Suppose that $A$ and $B$ are $m\times n$ matrices. Then $\transpose{(A+B)}=\transpose{A}+\transpose{B}$.

Theorem TMSM (Transpose and Matrix Scalar Multiplication) Suppose that $\alpha\in\complex{\null}$ and $A$ is an $m\times n$ matrix. Then $\transpose{(\alpha A)}=\alpha\transpose{A}$.

Theorem TT (Transpose of a Transpose) Suppose that $A$ is an $m\times n$ matrix. Then $\transpose{\left(\transpose{A}\right)}=A$.

Matrices and Complex Conjugation

As we did with vectors (Definition CCCV), we can define what it means to take the conjugate of a matrix.

Definition CCM (Complex Conjugate of a Matrix) Suppose $A$ is an $m\times n$ matrix. Then the conjugate of $A$, written $\conjugate{A}$ is an $m\times n$ matrix defined by \begin{equation*} \matrixentry{\conjugate{A}}{ij}=\conjugate{\matrixentry{A}{ij}} \end{equation*}

Example CCM: Complex conjugate of a matrix.

The interplay between the conjugate of a matrix and the two operations on matrices is what you might expect.

Theorem CRMA (Conjugation Respects Matrix Addition) Suppose that $A$ and $B$ are $m\times n$ matrices. Then $\conjugate{A+B}=\conjugate{A}+\conjugate{B}$.

Theorem CRMSM (Conjugation Respects Matrix Scalar Multiplication) Suppose that $\alpha\in\complex{\null}$ and $A$ is an $m\times n$ matrix. Then $\conjugate{\alpha A}=\conjugate{\alpha}\conjugate{A}$.

Theorem CCM (Conjugate of the Conjugate of a Matrix) Suppose that $A$ is an $m\times n$ matrix. Then $\conjugate{\left(\conjugate{A}\right)}=A$.

Finally, we will need the following result about matrix conjugation and transposes later.

Theorem MCT (Matrix Conjugation and Transposes) Suppose that $A$ is an $m\times n$ matrix. Then $\conjugate{\left(\transpose{A}\right)}=\transpose{\left(\conjugate{A}\right)}$.

The combination of transposing and conjugating a matrix will be important in subsequent sections, such as Subsection MINM.UM:Matrix Inverses and Nonsingular Matrices: Unitary Matrices and Section OD:Orthonormal Diagonalization. We make a key definition here and prove some basic results in the same spirit as those above.

Definition A (Adjoint) If $A$ is a matrix, then its adjoint is $\adjoint{A}=\transpose{\left(\conjugate{A}\right)}$.

You will see the adjoint written elsewhere variously as $A^H$, $A^\ast$ or $A^\dagger$. Notice that Theorem MCT says it does not really matter if we conjugate and then transpose, or transpose and then conjugate.

Theorem AMA (Adjoint and Matrix Addition) Suppose $A$ and $B$ are matrices of the same size. Then $\adjoint{\left(A+B\right)}=\adjoint{A}+\adjoint{B}$.

Theorem AMSM (Adjoint and Matrix Scalar Multiplication) Suppose $\alpha\in\complexes$ is a scalar and $A$ is a matrix. Then $\adjoint{\left(\alpha A\right)}=\conjugate{\alpha}\adjoint{A}$.

Theorem AA (Adjoint of an Adjoint) Suppose that $A$ is a matrix. Then $\adjoint{\left(\adjoint{A}\right)}=A$

Take note of how the theorems in this section, while simple, build on earlier theorems and definitions and never descend to the level of entry-by-entry proofs based on Definition ME. In other words, the equal signs that appear in the previous proofs are equalities of matrices, not scalars (which is the opposite of a proof like that of Theorem TMA).