令 
為一個 
階矩陣。LU 分解是指將 表示為兩個 階三角矩陣的乘積
，
其中 
是下三角矩陣，
是上三角矩陣，如下例，
![\left[\!\!\begin{array}{rrc} 3&-1&2\\ 6&-1&5\\ -9&7&3 \end{array}\!\!\right]=\left[\!\!\begin{array}{rcc} 1&0&0\\ 2&1&0\\ -3&4&1 \end{array}\!\!\right]\left[\!\!\begin{array}{crc} 3&-1&2\\ 0&1&1\\ 0&0&5 \end{array}\!\!\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5C%21%5C%21%5Cbegin%7Barray%7D%7Brrc%7D++3%26-1%262%5C%5C++6%26-1%265%5C%5C++-9%267%263++%5Cend%7Barray%7D%5C%21%5C%21%5Cright%5D%3D%5Cleft%5B%5C%21%5C%21%5Cbegin%7Barray%7D%7Brcc%7D++++1%260%260%5C%5C++2%261%260%5C%5C++-3%264%261++%5Cend%7Barray%7D%5C%21%5C%21%5Cright%5D%5Cleft%5B%5C%21%5C%21%5Cbegin%7Barray%7D%7Bcrc%7D++3%26-1%262%5C%5C++0%261%261%5C%5C++0%260%265++%5Cend%7Barray%7D%5C%21%5C%21%5Cright%5D&bg=ffffff&fg=000000&s=0 "\left[\!\!\begin{array}{rrc} 3&-1&2\\ 6&-1&5\\ -9&7&3 \end{array}\!\!\right]=\left[\!\!\begin{array}{rcc} 1&0&0\\ 2&1&0\\ -3&4&1 \end{array}\!\!\right]\left[\!\!\begin{array}{crc} 3&-1&2\\ 0&1&1\\ 0&0&5 \end{array}\!\!\right]")
。
LU 分解的本質是高斯消去法的一種表達形式，矩陣 記錄消去法化簡 的過程，而矩陣 則儲存化簡結果 (見“高斯消去法”)。LU 分解的外表看似平淡無奇，但它可以用來解線性方程，逆矩陣和計算行列式，堪稱是最具實用價值的矩陣分解式之一。
源自於 http://www.gregthatcher.com/Mathematics/LU_Factorization.aspx
This Calculator will Factorize a Square Matrix into the form A=LU where L is a lower triangular matrix, and U is an upper triangular matrix.
Add (2/3 * row1) to row2
(the
elementary matrix, hilighted in yellow, performs this operation for us.)
Add (-1/2 * row1) to row3
(the
elementary matrix, hilighted in yellow, performs this operation for us.)
Add (3/8 * row2) to row3
(the
elementary matrix, hilighted in yellow, performs this operation for us.)
Let's
multiply both sides of the equation by this
inverse.
We can simplify this to:
Let's
multiply both sides of the equation by this
inverse.
We can simplify this to:
Let's
multiply both sides of the equation by this
inverse.
We can simplify this to:
and we have now factored the matrix into the form A=LU.
Note that the 'L' matrix shows us the operations we performed to reduce the matrix (with signs changed).
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