Basic operations on matrices行列ぎょうれつ

date2026-07-14document_iddoc_c906c7cb97b9478908ef5d463658ca7ddescription行列をサイズをもつ数表として導入し、和・差・スカラー倍・行ベクトルと列ベクトルの区別を整理する講義である。prerequisitesベクトルの成分表示 / 数の四則演算type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecture

1Introduction

A matrix行列ぎょうれつ has a number of rows and a number of columns. Matrix operations are defined only when their size conditions are satisfied. This size discipline comes before interpreting matrices as linear maps.

2Terms and definitions

A matrix is a rectangular array of numbers arranged in rows行ぎょう and columns列れつ. A matrix with $m$ rows and $n$ columns is an $m \times n$ matrix.

For example,

A = (\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{matrix})

is a $3 \times 2$ matrix. A row vector行ぎょうベクトル is a matrix with one row, and a column vector列れつベクトル is a matrix with one column.

3Strategy

For matrix operations, first check the size. Sums and differences are defined only for matrices of the same size, while scalar multiplication is defined for matrices of any size.

data/lecture/math/linear-algebra/vector-operations.lecture.n.md

4Intuitive explanation

Adding matrices means adding entries in the same position. Like matching the same seat in two seating charts, only entries with the same row number and column number are paired.

Therefore matrices of different sizes do not have matching positions everywhere. This is why the operation is not defined.

5Addition and scalar multiplication

Let $A = (a_{i j})$ and $B = (b_{i j})$ be matrices of the same size $m \times n$ . Then

A + B = (a_{i j} + b_{i j}), A - B = (a_{i j} - b_{i j}) .

For a scalar $c$ ,

c A = (c a_{i j}) .

Two matrices are equal only when they have the same size and all corresponding entries are equal.

6Worked example

Let

A = (\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}), B = (\begin{matrix} 5 & 0 \\ - 1 & 2 \end{matrix}) .

Then

A + B = (\begin{matrix} 6 & 2 \\ 2 & 6 \end{matrix}), 2 A = (\begin{matrix} 2 & 4 \\ 6 & 8 \end{matrix}) .

A $2 \times 2$ matrix and a $2 \times 3$ matrix cannot be added because their positions do not correspond.

7Proof checkpoint: why same size is required

The entry $(A + B)_{i j}$ is defined by pairing the $(i, j)$ entry of $A$ with the $(i, j)$ entry of $B$ . If one matrix lacks that position, the rule has no value to assign. The restriction is therefore part of the definition, not a cosmetic convention.

8Common misunderstandings

Do not add matrices of different sizes because their entries look similar.
A row vector行ぎょうベクトル and a column vector列れつベクトル with the same number of components still have different sizes.
Square matrices are not the only matrices used in linear algebra; rectangular matrices are central for maps $K^{n} \to K^{m}$ .

9Scope

Addition and scalar multiplication are entrywise. Matrix multiplication is not entrywise and needs its own definition.

10Final forms

[PARSE ERROR: Undefined("Command(\"boxed\")")] A + B = (a_{i j} + b_{i j}) whensizesmatch

[PARSE ERROR: Undefined("Command(\"boxed\")")] c A = (c a_{i j})

[PARSE ERROR: Undefined("Command(\"boxed\")")] Alwayscheckmatrixsizesfirst [PARSE ERROR: Undefined("RBrace")]

11Exercise links

data/exercise/math/linear-algebra/matrix-computation-and-linear-transformations.exercise.n.md