Matrix Analysis
In
mathematics, particularly in linear algebra and applications, matrix analysis
is the study of matrices and their algebraic properties.
Some
particular topics out of many include; operations defined on matrices (such as
matrix addition, matrix multiplication and operations derived from these),
functions of matrices (such as matrix exponentiation and matrix logarithm, and
even sines and cosines etc. of matrices) and the eigenvalues of matrices
(eigendecomposition of a matrix, eigenvalue perturbation theory).
The
set of all m×n matrices over a number field F denoted in this article Mmn(F)
form a vector space. Examples of F include the set of integers ℤ,
the real numbers ℝ, and set of complex
numbers ℂ.
The spaces Mmn(F) and Mpq(F) are different spaces if m and p are unequal, and
if n and q are unequal; for instance M32(F) ≠ M23(F). Two m×n matrices A and B
in Mmn(F) can be added together to form another matrix in the space Mmn(F):
A , B ∈
M m n ( F ) , A + B ∈
M m n ( F ) {\displaystyle \mathbf {A} ,\mathbf {B} \in M_{mn}(F)\,,\quad
\mathbf {A} +\mathbf {B} \in M_{mn}(F)} \mathbf{A},\mathbf{B} \in
M_{mn}(F)\,,\quad \mathbf{A} + \mathbf{B} \in M_{mn}(F)
and
multiplied by a α in F, to obtain another matrix in Mmn(F):
α ∈
F , α A ∈ M m n ( F ) {\displaystyle
\alpha \in F\,,\quad \alpha \mathbf {A} \in M_{mn}(F)} \alpha \in F \,,\quad \alpha
\mathbf{A} \in M_{mn}(F)
Combining
these two properties, a linear combination of matrices A and B are in Mmn(F) is
another matrix in Mmn(F):
α A + β B ∈ M m n ( F ) {\displaystyle \alpha
\mathbf {A} +\beta \mathbf {B} \in M_{mn}(F)} \alpha \mathbf{A} +
\beta\mathbf{B} \in M_{mn}(F)
where
α and β are numbers in F.
Any
matrix can be expressed as a linear combination of basis matrices, which play
the role of the basis vectors for the matrix space. For example, for the set of
2×2 matrices over the field of real numbers, M22(ℝ),
one legitimate basis set of matrices is:
( 1 0 0 0 ) , ( 0 1 0 0 ) , ( 0 0 1 0 ) , (
0 0 0 1 ) , {\displaystyle {\begin{pmatrix}1&0\\0&0\end{pmatrix}}\,,\quad
{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\,,\quad
{\begin{pmatrix}0&0\\1&0\end{pmatrix}}\,,\quad
{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,,}
\begin{pmatrix}1&0\\0&0\end{pmatrix}\,,\quad \begin{pmatrix}0&1\\0&0\end{pmatrix}\,,\quad
\begin{pmatrix}0&0\\1&0\end{pmatrix}\,,\quad \begin{pmatrix}0&0\\0&1\end{pmatrix}\,,
because
any 2×2 matrix can be expressed as:
(
a b c d ) = a ( 1 0 0 0 ) + b ( 0 1 0 0 )
+ c ( 0 0 1 0 ) + d ( 0 0 0 1 ) , {\displaystyle
{\begin{pmatrix}a&b\\c&d\end{pmatrix}}=a{\begin{pmatrix}1&0\\0&0\end{pmatrix}}+b{\begin{pmatrix}0&1\\0&0\end{pmatrix}}+c{\begin{pmatrix}0&0\\1&0\end{pmatrix}}+d{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,,}
\begin{pmatrix}a&b\\c&d\end{pmatrix}=a
\begin{pmatrix}1&0\\0&0\end{pmatrix}
+b\begin{pmatrix}0&1\\0&0\end{pmatrix}
+c\begin{pmatrix}0&0\\1&0\end{pmatrix}
+d\begin{pmatrix}0&0\\0&1\end{pmatrix}\,,
where
a, b, c,d are all real numbers. This idea applies to other fields and matrices
of higher dimensions.
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