Linear Algebra and Its Applications, Exercise 1.7.1

Exercise 1.7.1. Find the LDU factorization for the matrix

$A = \begin{bmatrix} 1&-1&&& \\ -1&2&-1&& \\ &-1&2&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix}$

(This is the same matrix as in equation (6) of section 1.7, only with $a_{11} = 1$ instead of $a_{11} = 2$.)

Answer: For the first step of elimination we multiply the first row times the multiplier $l_{21} = -1$ and subtract it from the second row:

$\begin{bmatrix} 1&-1&&& \\ -1&2&-1&& \\ &-1&2&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix} \rightarrow \begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &-1&2&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix}$

In the second step we multiply the second row by the multiplier $l_{32} = -1$ and subtract it from the third row:

$\begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &-1&2&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix} \rightarrow \begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix}$

In the third step we multiply the third row by the multiplier $l_{43} = -1$ and subtract it from the fourth row:

$\begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix} \rightarrow \begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&&1&-1 \\ &&&-1&2 \end{bmatrix}$

Finally we we multiply the fourth row by the multiplier $l_{54} = -1$ and subtract it from the fifth row:

$\begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&&1&-1 \\ &&&-1&2 \end{bmatrix} \rightarrow \begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&&1&-1 \\ &&&&1 \end{bmatrix}$

Note that all steps feature the same multiplier. The final factorization is

$A = LDU = \begin{bmatrix} &&&& \\ -1&&&& \\ &-1&&& \\ &&-1&& \\ &&&-1& \end{bmatrix} \begin{bmatrix} 1&&&& \\ &1&&& \\ &&1&& \\ &&&1& \\ &&&&1 \end{bmatrix} \begin{bmatrix} &-1&&& \\ &&-1&& \\ &&&-1& \\ &&&&-1 \\ &&&& \end{bmatrix}$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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