3.1: Matrices

Introduction¶

The easiest way to think of a matrix is as being a collection of vectors, written next to each other, in one big array of numbers. Suppose we have three vectors in $\mathbb R^2$:

Place them side by side, keeping their order:

Shape¶

Here $A$ has two rows and three columns, so $A\in\mathbb R^{2\times3}$.

Here are several more matrices, with their shape underneath:

A square matrix has the same number of rows and columns. Square matrices are special in many ways, and we will spend time discussing them in detail.

Entries¶

We use capital letters such as $A$ and $B$ for matrices. The notation $a_{ij}$ denotes the entry of $A$ in row $i$, column $j$, counting from 1. In our original matrix,

For an arbitrary $n\times d$ matrix,

Two matrices are equal if they have the same shape and equal entries at every corresponding position. Merely containing the same numbers is not enough.

Rows and columns as vectors¶

Return to

There are two groups of vectors embedded in this matrix:

• The columns make up three vectors in $\mathbb R^2$:

  $$\begin{bmatrix}2\\1\end{bmatrix},\qquad \begin{bmatrix}-1\\4\end{bmatrix},\qquad \begin{bmatrix}3\\0\end{bmatrix}.$$

• The rows make up two vectors in $\mathbb R^3$:

  $$\begin{bmatrix}2\\-1\\3\end{bmatrix},\qquad \begin{bmatrix}1\\4\\0\end{bmatrix}.$$

These are two ways to view the same matrix.

More generally, an $n\times d$ matrix contains $d$ columns, viewed as vectors in $\mathbb R^n$, and $n$ rows, viewed as vectors in $\mathbb R^d$.

A subtlety is that these are different objects:

The first is a $3\times1$ matrix, which we call a vector in $\mathbb R^3$. The second is a $1\times3$ matrix, which we might call a row vector. In these notes, $\mathbb R^n$ refers to column vectors. When we discuss the vectors formed by a matrix’s rows, we write their entries vertically, as above.

Viewing a matrix as a collection of vectors lets us ask familiar questions:

• What is the span of the columns of $A$? These columns are vectors in $\mathbb R^2$, so their span is a subset of $\mathbb R^2$.

• Are the rows of $A$ linearly independent? Here we ask whether the two vectors in $\mathbb R^3$ formed from the rows are linearly independent.

Example: A dataset¶

Suppose we survey five Math 124 students about their past week. For each student, we record hours spent studying, practice problems completed, and lectures attended. The following numbers are made up:

Store the numerical values in a matrix, keeping the row and column order:

With one row per person and one column per feature (or variable), the two views have concrete meanings:

• Each of the five rows is a vector in $\mathbb R^3$: three measurements about one student.

• Each of the three columns is a vector in $\mathbb R^5$: one measurement recorded across all five students. For example, the first column contains everyone’s study hours.

We could instead choose one row per feature and one column per person:

Either arrangement works. We just need to say what the rows and columns represent.

Addition and scalar multiplication¶

Addition and scalar multiplication work just as they did for vectors.

For example,

Both inputs have shape $2\times3$, and so does their sum. A $2\times3$ matrix and a $3\times2$ matrix cannot be added, even though both contain six entries.

For example,

We define subtraction by $A-B=A+(-1)B$. The zero matrix $0_{n\times d}$ has $n$ rows, $d$ columns, and every entry equal to zero; $A+0_{n\times d}=A$ for any $n\times d$ matrix $A$.