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'partitioned matrix'
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| Title of object: |
partitioned matrix |
| Canonical Name: |
PartitionedMatrix |
| Type: |
Definition |
| Created on: |
2003-04-04 17:27:04 |
| Modified on: |
2006-07-07 11:43:00 |
| Classification: |
msc:15-00 |
| Keywords: |
Jordan canonical form, rational canonical form, smith normal form |
| Defines: |
block matrix, sub-matrix |
Revision comment (for changes between this and next version):
| Changes for correction #9693 ('also defines submatrix'). |
Preamble:
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Content:
A \emph{partitioned matrix}, or a \emph{block matrix}, is
a matrix $M$ that has been constructed from other smaller matrices.
These smaller matrices are called \emph{blocks} or \emph{sub-matrices}
of $M$.
For instance, if we partition the below $5\times 5$ matrix
as follows
\begin{eqnarray*}
L&=&\left( \begin{array}{cc|ccc}
1 & 0 & 1 & 2 & 3 \\
0 & 1 & 1 & 2 & 3 \\
\hline
2 & 3 & 9 & 9 & 9 \\
2 & 3 & 9 & 9 & 9 \\
2 & 3 & 9 & 9 & 9 \\
\end{array} \right),
\end{eqnarray*}
then we can define the matrices
\begin{equation*}
A=\left( \begin{array}{cc}
1 & 0 \\
0 & 1
\end{array} \right),
B=\left( \begin{array}{ccc}
1 & 2 & 3\\
1 & 2 & 3
\end{array} \right),
C=\left( \begin{array}{cc}
2 & 3 \\
2 & 3 \\
2 & 3
\end{array} \right),
D=\left( \begin{array}{ccc}
9 & 9 & 9 \\
9 & 9 & 9 \\
9 & 9 & 9 \\
\end{array} \right)
\end{equation*}
and write $L$ as
\begin{equation*}
L=\left( \begin{array}{cc}
A & B \\
C & D
\end{array} \right),\, \mbox{or\,\,}
L=\left( \begin{array}{c|c}
A & B \\
\hline
C & D
\end{array} \right).
\end{equation*}
If $A_1,\ldots, A_n$ are square matrices (of possibly
different sizes), then we define the \emph{direct sum} of
the matrices $A_1,\ldots, A_n$
as the partitioned matrix
$$\operatorname{diag}(A_1,\ldots, A_n) =\left( \begin{array}{c|c|c}
A_1 & & \\
\hline
& \ddots & \\
\hline
& & A_n \\
\end{array} \right),$$
where the off-diagonal blocks are zero.
If $A$ and $B$ are matrices of the same size partitioned into blocks of the same size, the partition of the sum is the sum of the partitions.
If $A$ and $B$ are $m\times n$ and $n\times k$ matrices, respectively, then if the blocks of $A$ and $B$ are of the correct size to be multiplied, then the blocks of the product are the products of the blocks. |
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