| A
lower triangular matrix is a square
matrix in which all elements above the main diagonal are zeros. |
| Square
matrices for which aij
= aji
are called symmetric about the main
diagonal. |
| Examples
of square matrices |
| |
The
diagonal matrix of order 4, |
|
The
identity matrix of order 3, |
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The
upper triangular matrix of order 4, |
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The
lower triangular matrix of order 4, |
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The
symmetric matrix of order 5, |
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The
elements of the main diagonal are in bold. |
|
| A
matrix whose all elements are zero is called the null
matrix, written 0. |
|
| Vectors |
| A
matrix with only one row or one column is called a vector. |
| A
matrix with one row and n
columns is called a row vector. |
| A
matrix with one column and m
rows is called a column vector. |
|
| Transposition |
| The
transpose is the matrix derived from a given matrix by
interchanging the rows and columns. Thus, the |
|
transpose of a matrix A
of order
m
´
n
is another matrix
denoted AT
of order n
´
m. |
| Therefore,
to transpose means to interchange the rows and columns of a
matrix, that
is, |
| AT
= [aji],
where j
is the column and i
is the row of matrix A
= [aij]. |
| Example:
Given a matrix A
of order 3 ´
4, find the transpose AT.
|
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|
| Basic
matrix
operations |
| Matrix addition and
subtraction |
| Matrices
can be added or subtracted only if they have the same
dimensions. |
| Addition |
| The
elements of the sum of the two matrices with the same
dimensions, A
and B,
are equal to the sums of |
| the
corresponding elements aij
+ bij
that is, |
| A
+ B
= [aij
+ bij]. |
| Subtraction |
| The
elements of the difference of the two matrices with the same
dimensions, A
and B,
are equal to the |
| differences
of the corresponding elements aij
-
bij
that is, |
| A
-
B
= [aij
-
bij]. |
| Example:
Given are matrices, A
and B, find the
sum A
+ B
and the difference A
-
B.
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| |
 |
and |
 |
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| |
then, |
 |
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and |
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| Scalar
multiplication of a matrix |
| Multiplication
of a matrix A
by a scalar c
is defined as |
| c
· A = [c · aij], |
| that
is, each element of the matrix is multiplied by c.
Therefore, if c
= 0, the result is the null
matrix. |
| Example:
Given the matrix |
 |
find the
product -
2 · A. |
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 |
|
| Matrix
multiplication |
| Multiplication
of two matrices is defined only if the number of columns of the
first matrix equals the number of |
| rows
of the second. Matrices that satisfy this condition are said to
be conformable. |
| If
A
= [aij]
is a matrix of order m
´
n
and B
= [bjk]
is a matrix of order n
´
p,
then the product |
| C
= AB |
| where
C
= [cik]
is the m
´
p
matrix of which the ikth
entries are defined by the formula |
 |
| Therefore, the
ikth
entry of the product equals the inner product of the components
of the ith
row of the first |
| matrix
with the components of the kth
column of the
second matrix. |
| Example:
Given are conformable matrices, A
and B, find the
product AB.
|
| |
 |
and |
 |
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|
| then, |
 |
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| or |
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| The
matrix-vector product |
| If
A = [aij]
is an m
´
n
matrix, x =
{xj}
a column vector of order n,
and y = {yi}
a column vector of order m |
| then,
y = Ax
is the matrix-vector product, |
| where |
 |
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|
| The
column dimension of the matrix A
must equal the row dimension of the vector x
to obey the product |
| definition. |
| Example:
Given is the matrix A
of order 2
´
3
and the column vector x
of order 3, find the
product Ax.
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| |
 |
and |
 |
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|
| then, |
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| If A is
a square matrix of order n
then,
|
| AI =
IA = A |
| where
I is the
identity
matrix of order n. |
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Matrices, determinants and systems of linear equations
contents
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© 2004 - 2012, Nabla Ltd. All rights reserved.
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