Matrices

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

Last updates: 12 June 2011

Matrices

Let $m$ and $n$ be positive integers. Let $R$ be a 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 $a$ .
$a = (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}) = (a_{i j})$

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 .

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.

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