Basics of the identity matrix単位行列たんいぎょうれつ, zero matrix零行列れいぎょうれつ, and transpose転置てんち

date2026-07-14document_iddoc_f5538408e41aa7dea09f22bd7c1e6c18description単位行列・零行列・転置を、行列演算の基準となる特別な行列と操作として整理し、転置が保存するもの・入れ替えるもの・行基本変形や列基本変形との対応まで説明する講義である。prerequisites行列のサイズ / 行列の基本演算type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecture

1Introduction

The identity matrix単位行列たんいぎょうれつ, the zero matrix零行列れいぎょうれつ, and the transpose転置てんち are reference objects and operations that should be clarified before more complicated matrix calculations.

The identity matrix represents doing nothing, the zero matrix collapses everything to the origin, and transpose exchanges the roles of rows and columns. Keeping these three roles separate makes inverse matrices, orthogonal matrices, symmetric matrices, and Hermitian matrices easier to read.

2Terms and definitions

The $n$ by $n$ identity matrix $I_{n}$ is the square matrix whose diagonal entries are $1$ and whose other entries are $0$ .

I_{n} = (\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix}) .

The zero matrix $O_{m, n}$ is the $m x n$ matrix whose entries are all $0$ . When the size is clear from context, we write simply $O$ .

For an $m x n$ matrix $A = (a_{i j})$ , the transpose $A^{T}$ is the $n x m$ matrix obtained by moving the entry in row $i$ , column $j$ to row $j$ , column $i$ .

(A^{T})_{i j} = a_{j i} .

3Intuitive explanation

The identity matrix is the linear map that returns every vector unchanged.

I_{n} v = v .

The zero matrix is the linear map that sends every vector to the zero vector.

O v = 0.

Transpose does not discard information. It rearranges entries like a reflection across the main diagonal, turning information that was read as rows into information that can be read as columns. Therefore transpose is important when relating the row space行空間ぎょうくうかん and the column space列空間れつくうかん.

4Basic laws

When the sizes are compatible, the following identities hold.

A I_{n} = A, I_{m} A = A .

A + O = A .

(A^{T})^{T} = A .

(A + B)^{T} = A^{T} + B^{T} .

(c A)^{T} = c A^{T} .

(A B)^{T} = B^{T} A^{T} .

The last identity reverses the order. This happens because the entries of $A B$ are made from rows of $A$ and columns of $B$ ; after transposing, those roles are exchanged.

5What transpose changes and preserves

Transpose changes the number of rows and columns, the way row space and column space are viewed, and the order of multiplication. An $m x n$ matrix becomes an $n x m$ matrix.

Transpose also preserves several quantities. It preserves the total number of entries, and for square matrices it preserves the determinant.

\det (A^{T}) = \det (A) .

It also preserves rank.

rank (A^{T}) = rank (A) .

The determinant and rank identities are previews proved in later lectures. The main point needed in this page is that transpose exchanges rows and columns and reverses the order of products.

For this reason, many statements about rows can be transposed into statements about columns.

6Transpose and elementary row/column operations

An elementary row operation行基本変形ぎょうきほんへんけい is represented by multiplying by an elementary matrix from the left. An elementary column operation列基本変形れつきほんへんけい is represented by multiplying by an elementary matrix from the right.

After transposing, an operation from the left becomes an operation from the right.

(E A)^{T} = A^{T} E^{T} .

Thus properties of row operations correspond to properties of column operations through transpose. This correspondence explains why row operations and column operations follow similar rules in determinant computations.

7Worked example: checking the reversed order under transpose

7.1Problem

For

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

verify $(A B)^{T} = B^{T} A^{T}$ .

7.2Explanation

First,

A B = (\begin{matrix} 2 & - 1 \\ 26 & 16 \end{matrix}),

so

(A B)^{T} = (\begin{matrix} 2 & 26 \\ - 1 & 16 \end{matrix}) .

On the other hand,

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

and therefore

B^{T} A^{T} = (\begin{matrix} 2 & 26 \\ - 1 & 16 \end{matrix}) .

The two results agree. This example is meant to make the reversal of order under transpose visible with concrete numbers.

8Common misunderstandings

The symbol $O$ has a size. The zero matrix in $A + O$ and the zero matrix in $A O$ may require different sizes.

The symbol $I$ also has a size. In $A I_{n} = A$ and $I_{m} A = A$ , the identity matrices on the two sides may have different sizes.

Transpose is not the inverse. In general, $A^{T} A = I$ is not true. This identity holds only for special matrices such as orthogonal or unitary matrices.

9Final form

The identity matrix does nothing, the zero matrix sends everything to zero, and transpose exchanges rows and columns.

I_{n} v = v, O v = 0, (A^{T})_{i j} = a_{j i} .

The especially important fact about transpose is that it reverses the order of multiplication.

(A B)^{T} = B^{T} A^{T} .

10Exercise links

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