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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