In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

##### DefinitionVSMVector 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.

##### DefinitionMEMatrix Equality

The $m\times n$ matrices $A$ and $B$ are equal, written $A=B$ provided $\matrixentry{A}{ij}=\matrixentry{B}{ij}$ for all $1\leq i\leq m\text{,}$ $1\leq j\leq n\text{.}$

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 $M_{mn}\text{.}$ Again, we will overload a symbol (`+') and a convention (juxtaposition for scalar multiplication).

Given the $m\times n$ matrices $A$ and $B\text{,}$ define the sum of $A$ and $B$ as an $m\times n$ matrix, written $A+B\text{,}$ by $\matrixentry{A+B}{ij}=\matrixentry{A}{ij}+\matrixentry{B}{ij}\text{,}$ for $1\leq i\leq m,\,1\leq j\leq n\text{.}$

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 does not relieve us from the obligation to state it carefully.

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.

##### DefinitionMSMMatrix Scalar Multiplication

Given the $m\times n$ matrix $A$ and the scalar $\alpha\in\complexes\text{,}$ the scalar multiple of $A$ is the $m\times n$ matrix, written $\alpha A\text{,}$ and defined by $\matrixentry{\alpha A}{ij}=\alpha\matrixentry{A}{ij}\text{,}$ for $1\leq i\leq m,\,1\leq j\leq n\text{.}$

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.

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.

##### Proof

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

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

##### DefinitionZMZero Matrix

The $m\times n$ zero matrix is written as $\zeromatrix=\zeromatrix_{m\times n}$ and defined by $\matrixentry{\zeromatrix}{ij}=0\text{,}$ for all $1\leq i\leq m\text{,}$ $1\leq j\leq n\text{.}$

# SubsectionTSMTransposes and Symmetric Matrices¶ permalink

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.

##### DefinitionTMTranspose of a Matrix

Given an $m\times n$ matrix $A\text{,}$ its transpose is the $n\times m$ matrix $\transpose{A}$ given by $\matrixentry{\transpose{A}}{ij}=\matrixentry{A}{ji}\text{,}$for $1\leq i\leq n,\,1\leq j\leq m\text{.}$

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.

##### DefinitionSYMSymmetric Matrix

The matrix $A$ is symmetric if $A=\transpose{A}\text{.}$

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

Before reading the next proof, we want to offer you some advice about how to become more proficient at constructing proofs. Perhaps you can apply this advice to the next theorem. Have a peek at Proof Technique P now.

##### Proof

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.

# SubsectionMCCMatrices and Complex Conjugation¶ permalink

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

##### DefinitionCCMComplex Conjugate of a Matrix

Suppose $A$ is an $m\times n$ matrix. Then the conjugate of $A\text{,}$ written $\conjugate{A}$ is an $m\times n$ matrix defined by $\matrixentry{\conjugate{A}}{ij}=\conjugate{\matrixentry{A}{ij}}$ for $1\leq i\leq m\text{,}$ $1\leq j\leq n\text{.}$

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

##### Proof

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

##### Proof

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

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

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

##### Proof

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).

##### 1

Perform the following matrix computation. \begin{equation*} (6) \begin{bmatrix} 2 & -2 & 8 & 1 \\ 4 & 5 & -1 & 3\\ 7 & -3 & 0 & 2 \end{bmatrix} + (-2) \begin{bmatrix} 2 & 7 & 1 & 2\\ 3 & -1 & 0 & 5\\ 1 & 7 & 3 & 3 \end{bmatrix} \end{equation*}

##### 2

Theorem VSPM reminds you of what previous theorem? How strong is the similarity?

##### 3

Compute the transpose of the matrix below. \begin{equation*} \begin{bmatrix} 6 & 8 & 4 \\ -2 & 1 & 0 \\ 9 & -5 & 6 \end{bmatrix} \end{equation*}

# SubsectionExercises

##### C10

Let$A = \begin{bmatrix} 1 & 4 & -3 \\ 6 & 3 & 0\end{bmatrix}\text{,}$ $B = \begin{bmatrix} 3 & 2 & 1 \\ -2 & -6 & 5\end{bmatrix}$ and $C = \begin{bmatrix} 2 & 4 \\ 4 & 0 \\ -2 & 2\end{bmatrix}\text{.}$ Let $\alpha = 4$ and $\beta = 1/2\text{.}$ Perform the following calculations: (1) $A + B\text{,}$ (2) $A + C\text{,}$ (3) $\transpose{B} + C\text{,}$ (4) $A + \transpose{B}\text{,}$ (5) $\beta C\text{,}$ (6) $4A - 3B\text{,}$ (7) $\transpose{A} + \alpha C\text{,}$ (8) $A + B - \transpose{C}\text{,}$ (9) $4A + 2B - 5\transpose{C}\text{.}$

Solution
##### C11

Solve the given vector equation for $x\text{,}$ or explain why no solution exists. \begin{equation*} 2 \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 2 \end{bmatrix} - 3 \begin{bmatrix} 1 & 1 & 2 \\ 0 & 1 & x \end{bmatrix} = \begin{bmatrix} -1 & 1 & 0 \\ 0 & 5 & -2 \end{bmatrix} \end{equation*}

Solution
##### C12

Solve the given matrix equation for $\alpha\text{,}$ or explain why no solution exists. \begin{equation*} \alpha\begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & -1 \end{bmatrix} + \begin{bmatrix} 4 & 3 & -6 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 12 & 6 \\ 6 & 4 & -2 \end{bmatrix} \end{equation*}

Solution
##### C13

Solve the given matrix equation for $\alpha\text{,}$ or explain why no solution exists. \begin{equation*} \alpha \begin{bmatrix} 3 & 1 \\ 2 & 0 \\ 1 & 4\end{bmatrix} - \begin{bmatrix} 4 & 1 \\ 3 & 2 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 1 & -2 \\ 2 & 6 \end{bmatrix} \end{equation*}

Solution
##### C14

Find $\alpha$ and $\beta$ that solve the following equation. \begin{equation*} \alpha\begin{bmatrix} 1 & 2 \\ 4 & 1 \end{bmatrix} + \beta\begin{bmatrix} 2 & 1 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 4 \\ 6 & 1 \end{bmatrix} \end{equation*}

Solution

In Chapter V we defined the operations of vector addition and vector scalar multiplication in Definition CVA and Definition CVSM. These two operations formed the underpinnings of the remainder of the chapter. We have now defined similar operations for matrices in Definition MA and Definition MSM. You will have noticed the resulting similarities between Theorem VSPCV and Theorem VSPM.

In Exercises M20–M25, you will be asked to extend these similarities to other fundamental definitions and concepts we first saw in Chapter V. This sequence of problems was suggested by Martin Jackson.

##### M20

Suppose $S=\set{B_1,\,B_2,\,B_3,\,\ldots,\,B_p}$ is a set of matrices from $M_{mn}\text{.}$ Formulate appropriate definitions for the following terms and give an example of the use of each.

1. A linear combination of elements of $S\text{.}$
2. A relation of linear dependence on $S\text{,}$ both trivial and nontrivial.
3. $S$ is a linearly independent set.
4. $\spn{S}\text{.}$
##### M21

Show that the set $S$ is linearly independent in $M_{22}\text{.}$ \begin{equation*} S=\set{ \begin{bmatrix}1&0\\0&0\end{bmatrix},\, \begin{bmatrix}0&1\\0&0\end{bmatrix},\, \begin{bmatrix}0&0\\1&0\end{bmatrix},\, \begin{bmatrix}0&0\\0&1\end{bmatrix} }\text{.} \end{equation*}

Solution
##### M22

Determine if the set $S$ below is linearly independent in $M_{23}\text{.}$ \begin{equation*} \set{ \begin{bmatrix} -2 & 3 & 4 \\ -1 & 3 & -2 \end{bmatrix},\, \begin{bmatrix} 4 & -2 & 2 \\ 0 & -1 & 1 \end{bmatrix},\, \begin{bmatrix}-1 & -2 & -2 \\ 2 & 2 & 2 \end{bmatrix},\, \begin{bmatrix}-1 & 1 & 0 \\ -1 & 0 & -2 \end{bmatrix},\, \begin{bmatrix}-1 & 2 & -2 \\ 0 & -1 & -2 \end{bmatrix} } \end{equation*}

Solution
##### M23

Determine if the matrix $A$ is in the span of $S\text{.}$ In other words, is $A\in\spn{S}\text{?}$ If so write $A$ as a linear combination of the elements of $S\text{.}$ \begin{align*} A&= \begin{bmatrix} -13 & 24 & 2\\ -8 & -2 & -20 \end{bmatrix}\\ S&=\set{ \begin{bmatrix} -2 & 3 & 4 \\ -1 & 3 & -2 \end{bmatrix},\, \begin{bmatrix} 4 & -2 & 2 \\ 0 & -1 & 1 \end{bmatrix},\, \begin{bmatrix}-1 & -2 & -2 \\ 2 & 2 & 2 \end{bmatrix},\, \begin{bmatrix}-1 & 1 & 0 \\ -1 & 0 & -2 \end{bmatrix},\, \begin{bmatrix}-1 & 2 & -2 \\ 0 & -1 & -2 \end{bmatrix} } \end{align*}

Solution
##### M24

Suppose $Y$ is the set of all $3\times 3$ symmetric matrices (Definition SYM). Find a set $T$ so that $T$ is linearly independent and $\spn{T}=Y\text{.}$

Solution
##### M25

Define a subset of $M_{33}$ by \begin{equation*} U_{33}=\setparts{ A\in M_{33} }{ \matrixentry{A}{ij}=0\text{ whenever }i\gt j }\text{.} \end{equation*} Find a set $R$ so that $R$ is linearly independent and $\spn{R}=U_{33}\text{.}$

##### T13

Prove Property CM of Theorem VSPM. Write your proof in the style of the proof of Property DSAM given in this section.

Solution

A matrix $A$ is skew-symmetric if $\transpose{A}=-A$ Exercises T30–T37 employ this definition.

##### T30

Prove that a skew-symmetric matrix is square. (Hint: study the proof of Theorem SMS.)

##### T31

Prove that a skew-symmetric matrix must have zeros for its diagonal elements. In other words, if $A$ is skew-symmetric of size $n\text{,}$ then $\matrixentry{A}{ii}=0$ for $1\leq i\leq n\text{.}$ (Hint: carefully construct an example of a $3\times 3$ skew-symmetric matrix before attempting a proof.)

##### T32

Prove that a matrix $A$ is both skew-symmetric and symmetric if and only if $A$ is the zero matrix. (Hint: one half of this proof is very easy, the other half takes a little more work.)

##### T33

Suppose $A$ and $B$ are both skew-symmetric matrices of the same size and $\alpha,\,\beta\in\complexes\text{.}$ Prove that $\alpha A + \beta B$ is a skew-symmetric matrix.

##### T34

Suppose $A$ is a square matrix. Prove that $A+\transpose{A}$ is a symmetric matrix.

##### T35

Suppose $A$ is a square matrix. Prove that $A-\transpose{A}$ is a skew-symmetric matrix.

##### T36

Suppose $A$ is a square matrix. Prove that there is a symmetric matrix $B$ and a skew-symmetric matrix $C$ such that $A=B+C\text{.}$ In other words, any square matrix can be decomposed into a symmetric matrix and a skew-symmetric matrix (Proof Technique DC). (Hint: consider building a proof on Exercise MO.T34 and Exercise MO.T35.)