## Linear Algebra and Its Applications, Review Exercise 1.17

Review exercise 1.17. Factor the following two symmetric matrices $A = \begin{bmatrix} 1&2&0 \\ 2&6&4 \\ 0&4&11 \end{bmatrix} \quad A = \begin{bmatrix} a&b \\ b&c \end{bmatrix}$

into the form $A = LDL^T$.

Answer: We start elimination for the first matrix by subtracting 2 times the first row from the second row ( $l_{21} = 2$): $\begin{bmatrix} 1&2&0 \\ 2&6&4 \\ 0&4&11 \end{bmatrix} \rightarrow \begin{bmatrix} 1&2&0 \\ 0&2&4 \\ 0&4&11 \end{bmatrix}$

and then subtract 2 times the second row from the third row ( $l_{32} = 2$): $\begin{bmatrix} 1&2&0 \\ 0&2&4 \\ 0&4&11 \end{bmatrix} \rightarrow \begin{bmatrix} 1&2&0 \\ 0&2&4 \\ 0&0&3 \end{bmatrix}$

We then have $L = \begin{bmatrix} 1&0&0 \\ 2&1&0 \\ 0&2&1 \end{bmatrix}$

and $U = \begin{bmatrix} 1&2&0 \\ 0&2&4 \\ 0&0&3 \end{bmatrix} = \begin{bmatrix} 1&0&0 \\ 0&2&0 \\ 0&0&3 \end{bmatrix} \begin{bmatrix} 1&2&0 \\ 0&1&2 \\ 0&0&1 \end{bmatrix} = DL^T$

We thus have $A = \begin{bmatrix} 1&0&0 \\ 2&1&0 \\ 0&2&1 \end{bmatrix} \begin{bmatrix} 1&0&0 \\ 0&2&0 \\ 0&0&3 \end{bmatrix} \begin{bmatrix} 1&2&0 \\ 0&1&2 \\ 0&0&1 \end{bmatrix} = LDL^T$

For the second matrix we subtract $b/a$ times the first row from the second row ( $l_{21} = b/a$): $\begin{bmatrix} a&b \\ b&c \end{bmatrix} \rightarrow \begin{bmatrix} a&b \\ 0&c - b^2/a \end{bmatrix}$

We then have $L = \begin{bmatrix} 1&0 \\ b/a&1 \end{bmatrix} \quad \rm \quad U = \begin{bmatrix} a&b \\ 0&c - b^2/a \end{bmatrix} = \begin{bmatrix} a&0 \\ 0&c - b^2/a \end{bmatrix} \begin{bmatrix} 1&b/a \\ 0&1 \end{bmatrix}$

and thus $A = \begin{bmatrix} 1&0 \\ b/a&1 \end{bmatrix} \begin{bmatrix} a&0 \\ 0&c - b^2/a \end{bmatrix} \begin{bmatrix} 1&b/a \\ 0&1 \end{bmatrix} = LDL^T$

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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