Matrices

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 14 September 2013

Matrices

Let $m$ and $n$ be positive integers. Let $R$ be a commutative ring.

An $m \times n$ matrix with entries in $R$ is a table of elements of $R$ with $m$ rows and $n$ columns.
A column vector of length $n$ is an $n \times 1$ matrix.
A row vector of length $n$ is an $1 \times n$ matrix.
The $(i, j)$ entry of a matrix $a$ is the element $a_{i j}$ of $R$ in row $i$ and column $j$ of $a$ .
$a = (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}) = (a_{i j})$

Let $M_{m \times n} (R)$ be the set of $m \times n$ matrices with entries in $R$ .

The sum of $m \times n$ matrices $a = (a_{r s})$ and $b = (b_{r s})$ is the $m \times n$ matrix $a + b$ given by
${(a + b)}_{i j} = a_{i j} + b_{i j}, for i = 1, 2, \dots, m, and j = 1, 2, \dots, n .$
Scalar multiplication of an element $r \in R$ with a matrix $a$ is the matrix $r a$ given by
${(r a)}_{i j} = r a_{i j}, for i = 1, 2, \dots, m, and j = 1, 2, \dots, n .$
The product of a $m \times n$ matrix $a = (a_{i j})$ and a $n \times p$ matrix $b = (b_{k ℓ})$ is the $m \times p$ matrix $a b$ given by
${(a b)}_{i j} = \sum_{k = 1}^{n} a_{i k} b_{k ℓ}, for i = 1, 2, \dots, m, and j = 1, 2, \dots, p .$
The transpose of a $m \times n$ matrix $a$ is the $n \times m$ matrix $a^{t}$ given by
${(a^{t})}_{i j} = a_{j i}, for i = 1, 2, \dots, n, and j = 1, 2, \dots, m .$

Example.

(\begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix}) + (\begin{matrix} 5 & 2 \\ 1 & 6 \end{matrix}) = (\begin{matrix} 7 & 5 \\ 2 & 8 \end{matrix}), (\begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix}) (\begin{matrix} 5 & 2 \\ 1 & 6 \end{matrix}) = (\begin{matrix} 13 & 22 \\ 7 & 14 \end{matrix}) and {(\begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix})}^{t} = (\begin{matrix} 2 & 1 \\ 3 & 2 \end{matrix}) .

HW: Show that if $a, b, c$ are $n \times n$ matrices then $(a b) c = a (b c)$ , $a (b + c) = a b + a c$ and $(b + c) a = b a + c a$ .

HW: Give an example of $n \times n$ matrices $a$ and $b$ such that $a b \neq b a$ .

Let $S$ be a set.

The Kronecker delta is given by
$δ_{a b} = {\begin{cases} 1, & if a, b \in S and a = b, \\ 0, & otherwise. \end{cases}$

Let

m

and

n

be positive integers and let

R

be a ring.

(a) The set

M_{m \times n} (R) = {m \times n matrices with entries in R},

with operation addition is an abelian group with zero element

0 \in M_{m \times n} (R)

given by

0_{i j} = 0 for i = 1, 2, \dots, m, and j = 1, 2, \dots, n .

(b) The set

M_{m \times n} (R)

with operations addition and scalar multiplication is an

R

-module.

M_{n} (R) = M_{n \times n} (R)

with operations addition and product is a ring with identity

1 \in M_{n} (R)

given by

1_{i j} = δ_{i j} for i = 1, 2, \dots, n, and j = 1, 2, \dots, n .

Proof.

(a)

To show:	$A + B = B + A$
To show:	${(A + B)}_{i j} = {(B + A)}_{i j}$
$\begin{matrix} {(A + B)}_{i j} & = & A_{i j} + B_{i j} \\ = & B_{i j} + A_{i j}, since 𝔽 has commutative addition \\ = & {(B + A)}_{i j} . \end{matrix}$

(b)

To show:	$(A + B) + C = A + (B + C)$
To show:	${((A + B) + C)}_{i j} = {(A + (B + C))}_{i j}$
$\begin{matrix} (A + B) + C = A + (B + C) & = & {(A + B)}_{i j} + C_{i j} = (A_{i j} + B_{i j}) + C_{i j} \\ = & A_{i j} + (B_{i j} + C_{i j}), since + is associative in 𝔽, \\ = & A_{i j} + {(B + C)}_{i j} = {(A + (B + C))}_{i j} . \end{matrix}$

(c)

Define the zero matrix

0

0_{i j} = 0 .

To show:

(ca)

0 + A = A

(cb)

A + 0 = A

(ca)

To show:	${(0 + A)}_{i j} = A_{i j}$
$\begin{matrix} {(0 + A)}_{i j} & = & 0_{i j} + A_{i j} = 0 + A_{i j} = A_{i j} . \end{matrix}$

(cb)

To show:	${(A + 0)}_{i j} = A_{i j}$
$\begin{matrix} {(A + 0)}_{i j} & = & A_{i j} + 0_{i j} = A_{i j} + 0 = A_{i j} . \end{matrix}$

(d)

To show:	$(A B) C = A (B C)$
To show:	${((A B) C)}_{i j} = {(A (B C))}_{i j}$
$\begin{matrix} {((A B) C)}_{i j} & = & \sum_{k = 1}^{n} {(A B)}_{i k} C_{k j} \\ = & \sum_{k = 1}^{n} \sum_{ℓ = 1}^{m} (A_{i ℓ} B_{ℓ k}) C_{k j} \\ = & \sum_{ℓ = 1}^{m} \sum_{k = 1}^{n} (A_{i ℓ} B_{ℓ k}) C_{k j} \\ = & \sum_{ℓ = 1}^{m} \sum_{k = 1}^{n} A_{i ℓ} (B_{ℓ k} C_{k j}) \\ = & \sum_{ℓ = 1}^{m} A_{i ℓ} {(B C)}_{ℓ j} . \end{matrix}$

(e)

To show:

(ea)

(A + B) \cdot C = A C + B C

(eb)

C (A + B) = C A + C B

(ea)

To show:	${((A + B) C)}_{i j} = {(A C + B C)}_{i j}$
$\begin{matrix} {((A + B) C)}_{i j} & = & \sum_{k = 1}^{n} {(A + B)}_{i k} C_{k j} = \sum_{k = 1}^{n} (A_{i k} B_{i k}) C_{k j} \\ = & \sum_{k = 1}^{n} (A_{i k} C_{k j} + B_{i k} C_{k j}), by the distributive property in 𝔽 \\ = & (\sum_{k = 1}^{n} A_{i k} C_{k j}) + (\sum_{k = 1}^{n} B_{i k} C_{k j}) \\ = & {(A C)}_{i j} + {(B C)}_{i j} \\ = & {(A C + B C)}_{i j} . \end{matrix}$

(eb)

To show:	${(C (A + B))}_{i j} = {(C A + C B)}_{i j}$
$\begin{matrix} {(C (A + B))}_{i j} & = & \sum_{k = 1}^{m} C_{i k} {(A + B)}_{k j} = \sum_{k = 1}^{m} C_{i k} (A_{k j} + B_{k j}) \\ = & \sum_{k = 1}^{m} (C_{i k} A_{k j} + C_{i k} B_{k j}), by the distributive property in 𝔽, \\ = & (\sum_{k = 1}^{m} C_{i k} A_{k j}) + (\sum_{k = 1}^{m} C_{i k} B_{k j}) \\ = & {(C A)}_{i j} + {(C B)}_{i j} \\ = & {(C A + C B)}_{i j} . \end{matrix}$

(f)

Define the identity matrix

1

1_{i j} = {\begin{matrix} 1, & if i = j, \\ 0, & if i \neq j . \end{matrix}

To show:

(fa)

1 \cdot A = A

(fb)

A \cdot 1 = A

(fa)

To show:	${(1 \cdot A)}_{i j} = A_{i j}$
$\begin{matrix} {(1 \cdot A)}_{i j} & = & \sum_{k = 1}^{n} 1_{i k} A_{k j} = 1_{i i} A_{i j} + \sum_{k \neq i} 1_{i k} A_{k j} \\ = & 1 \cdot A_{i j} + 0 = A_{i j} . \end{matrix}$

(fb)

To show:	${(A \cdot 1)}_{i j} = A_{i j}$
$\begin{matrix} {(A \cdot 1)}_{i j} & = & \sum_{k = 1}^{m} A_{i k} 1_{k j} = A_{i j} 1_{j j} + \sum_{k \neq j} A_{i k} 1_{k j} \\ = & A_{i j} \cdot 1 + 0 = A_{i j} . \end{matrix}$

(g)

Define the matrix

- A

{(- A)}_{i j} = - A_{i j} .

To show:

(ga)

A + (- A) = 0

(gb)

(- A) + A = 0

(ga)

To show:	${(A + (- A))}_{i j} = 0_{i j}$
$\begin{matrix} {(A + (- A))}_{i j} & = & A_{i j} + {(- A)}_{i j} = A_{i j} + (- A_{i j}) \\ = & 0 = 0_{i j} \end{matrix}$

(gb)

To show:	${((- A) + A)}_{i j} = 0_{i j}$
${((- A) + A)}_{i j} = {(- A)}_{i j} + A_{i j} = - A_{i j} + A_{i j} = 0 = 0_{i j} .$

$□$

HW: Show that $M_{n \times 1} (R) ≃ R^{n}$ as $R$ -modules.

Example.

(a) The set of triangular matrices and the set of diagonal matrices are subrings of

M_{n} (R)

(b) The ideals of

M_{n} (R)

are

M_{n} (I)

, where

I

is an ideal of

R

Let $a$ be an $n \times n$ matrix with entries $a_{i j}$ .

The trace of $a$ is
$tr (a) = \sum_{i = 1}^{n} a_{i i}$ .
The determinant of $a$ is
$\det (a) = \sum_{w \in S_{n}} \det (w) a_{1 w (1)} a_{2 w (2)} \dots a_{n w (n)}$

Let

𝔽

be a field and let

n \in ℤ_{> 0}

(a) Up to constant multiples,

tr : M_{n} (𝔽) \to 𝔽

is the unique function such that

(a1) If

c \in 𝔽

and

a, b \in M_{n} (𝔽)

then

tr (a + b) = tr (a) + tr (b) and tr (c a) = c tr (a) .

(a2) If

a, b \in M_{n} (𝔽) then

tr (a b) = tr (b a)

(b) Identify

M_{n} (𝔽)

with the

𝔽

-module

\underset{\underset{n times}{⏟}}{𝔽^{n} \times \dots \times 𝔽^{n}}

\begin{matrix} M_{n} (𝔽) & \tilde{⟶} & \underset{\underset{n times}{⏟}}{𝔽^{n} \times \dots \times 𝔽^{n}} \\ a & ⟼ & (a_{1} | a_{2} | \dots | a_{n}) \end{matrix}

, where

a_{i}

are the columns of

a

The function

\det : M_{n} (𝔽) \to 𝔽

is the unique function such that

(b1) (columnwise linear) If

i \in {1, 2, \dots, n}

and

c \in 𝔽

then

\det (a_{1} | \dots | a_{i} + b_{i} | \dots | a_{n}) = \det (a_{1} | \dots | a_{i} | \dots | a_{n}) + \det (a_{1} | \dots | b_{i} | \dots | a_{n})

and

\det (a_{1} | \dots | c a_{i} | \dots | a_{n}) = c \det (a_{1} | \dots | a_{i} | \dots | a_{n})

(b2) If

i \in {1, 2, \dots, n - 1}

then

\det (a_{1} | \dots | a_{i} | a_{i + 1} | \dots | a_{n}) = - \det (a_{1} | \dots | a_{i + 1} | a_{i} | \dots | a_{n})

(b3) if

a, b \in M_{n} (𝔽) then

\det (a b) = \det (a) \det (b)

Notes and References

A matrix is just a table of numbers, and hence matrices appear everywhere. These notes are taken from notes of Arun Ram from 1999. A nice solid reference is [HP].

References

[HP] W.V.D. Hodge and D. Pedoe, Methods of algebraic geometry. Vol. I. Reprint of the 1947 original. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1994. viii+440 pp. ISBN: 0-521-46900-7, 14-01 (01A75) Methods of algebraic geometry I, Cambrdige University Press, MR1288305.

page history