Triangular matrix: Difference between revisions

Revision as of 10:08, 9 March 2021

Revision as of 06:22, 16 March 2021

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\end{matrix}</math>

Observe that the first equation (<math>\ell_{1,1} x_1 = b_1</math>) only involves <math>x_1</math>, and thus one can solve for <math>x_1</math> directly. The second equation only involves direc<mathref>x_1</mathref> and <math>x_2</math>, and thus can be solved once one substitutes in the already solved value for <math>x_1</math>. Continuing in this way, the <math>k</math>-th equation only involves <math>x_1,\dotsts,x_k</math>, and one can solve for <math>x_k</math> using the previously solved values for <math>x_1,\dots,x_{k-1}</math>.

The resulting formulas are:

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x_m &= \frac{b_m - \sum_{i=1}^{m-1} \ell_{m,i}x_i}{\ell_{m,m}}.

\end{align}</math>

A matrix equation with an upper triangular matrix ''U'' can be solved in an analogous way, only working backwards.

===Applications===