A packed storage matrix, also known as packed matrix, is a term used in programming for representing an <math>m\times n</math> matrix. It is a more compact way than an m-by-n rectangular array by exploiting a special structure of the matrix.
Typical examples of matrices that can take advantage of packed storage include:
- symmetric or hermitian matrix
- Triangular matrix
- Banded matrix.
Triangular packed matrices
The packed storage matrix allows a matrix to be converted to an array, shrinking the matrix significantly. In doing so, a square <math>n \times n</math> matrix is converted to an array of length .
Consider the following upper matrix:
:<math>\mathbf{U} = \begin{pmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
& a_{22} & a_{23} & a_{24} \\
& & a_{33} & a_{34} \\
& & & a_{44} \\
\end{pmatrix}</math>
which can be packed into the one array:
:<math> \mathbf{UP} = (\underbrace{a_{11\ \underbrace{a_{12}\ a_{22\ \underbrace{a_{13}\ a_{23}\ a_{33\ \underbrace{a_{14},\ a_{24}\ a_{34}\ a_{44)
</math>
Similarly the lower matrix:
:<math>\mathbf{L} = \begin{pmatrix}
a_{11} & & & \\
a_{21} & a_{22} & & \\
a_{31} & a_{32} & a_{33} & \\
a_{41} & a_{42} & a_{43} & a_{44} \\
\end{pmatrix}.</math>
can be packed into the following one dimensional array:
:<math>
LP = (\underbrace{a_{11}\ a_{21}\ a_{31}\ a_{41\ \underbrace{a_{22}\ a_{32}\ a_{42\ \underbrace{a_{33}\ a_{43\ \underbrace{a_{44)
</math>