3.9 KiB
Matrix arithmetic
Definitions
Definition: let
A
be am \times n
matrix given by
A = \begin{pmatrix} a_{11} & a_{12}& \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}
with
a_{ij}
referred to as the entries ofA
or scalars in general, with(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}
. For real entries inA
we may denoteA \in \mathbb{R}^{m \times n}
.
This matrix may be denoted in a shorter way by A = (a_{ij})
.
Definition: let
\mathbf{x}
be a1 \times n
matrix, referred to as row vector given by
\mathbf{x} = \begin{pmatrix}x_1 \ x_2 \ \vdots \ x_n\end{pmatrix}
with
x_i
referred to as the entries of\mathbf{x}
, withi \in \{1, \dots, n\}
. For real entries we may denote\mathbf{x} \in \mathbb{R}^n
.
Definition: let
\mathbf{x}
be an \times 1
matrix, referred to as column vector given by
\mathbf{x} = (x_1, x_2, \dots, x_n)
with
x_i
referred to as the entries of\mathbf{x}
, withi \in \{1, \dots, n\}
. Also for the column vector we have for real entries\mathbf{x} \in \mathbb{R}^n
.
From these two definitions it may be observed that row and column vectors may be used interchangebly, however using both it is important to state the difference. Best practice is to always work with row vectors and take the transpose if necessary.
Matrix operations
Definition: two
m \times n
matricesA
andB
are said to be equal ifa_{ij} = b_{ij}
for eachi(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}
.
Definition: if
A
is anm \times n
matrix and\alpha
is a scalar, then\alpha A
is them \times n
matrix whose(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}
entry is\alpha a_{ij}
.
Definition: if
A = (a_{ij})
andB = (b_{ij})
are bothm \times n
matrices, then the sumA + B
is them \times n
matrix whose(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}
entry isa_{ij} + b_{ij}
for each ordered pair(i,j)
.
If A
is an m \times n
matrix and \mathbf{x}
is a vector in \mathbb{R}^n
, then
A \mathbf{x} = x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \dots + x_n \mathbf{a}_n
with A = (\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n})
.
Definition: if
\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n}
are vectors in\mathbb{R}^m
andx_1, x_2 \dots, x_n
are scalars, then a sum of the form
x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \dots + x_n \mathbf{a}_n
is said to be a linear combination of the vectors
\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n}
.
Theorem: a linear system
A \mathbf{x} = \mathbf{b}
is consistent if and only if\mathbf{b}
can be written as a linear combination of the column vectorsA
.
??? note "Proof:"
Will be added later.
Transpose matrix
Definition: the transpose of an
m \times n
matrix A is then \times m
matrixB
defined by
b_{ji} = a_{ij},
for
j \in \{1, \dots, n\}
andi \in \{1, \dots m\}
. The transpose ofA
is denoted byA^T
.
Definition: an
n \times n
matrixA
is said to be symmetric ifA^T = A
.
Hermitian matrix
Definition: the conjugate transpose of an
m \times n
matrix A is then \times m
matrixB
defined by
b_{ji} = \bar a_{ij},
for
j \in \{1, \dots, n\}
andi \in \{1, \dots m\}
. The conjugate transpose ofA
is denoted byA^H
.
Definition: an
n \times n
matrixA
is said to be Hermitian ifA^H = A
.
Matrix multiplication
Definition: if
A = (a_{ij})
is anm \times n
matrix andB = (b_{ij})
is ann \times r
matrix, then the productA B = C = (c_{ij})
is them \times r
matrix whose entries are defined by
c_{ij} = \mathbf{a}i \mathbf{b}j = \sum{k=1}^n a{ik} b_{kj}