A symmetric positive definite matrix, such as the mass matrix of a multidegree-of-freedom system, \([m]\), can be expressed as the product of a lower triangular matrix, \([L]\), and an upper triangular matrix, \([L]^{T}\), as

using a procedure known as the Choleski method [6.18]. For a mass matrix of order \(3 \times 3\), Eq. (E.1) becomes

By carrying out the multiplication of the matrices on the right-hand side of Eq. (E.2) and equating each of the elements of the resulting \(3 \times 3\) matrix to the corresponding element of the matrix on the left-hand side of Eq. (E.2), the matrix \([L]\) can be identified. Using this procedure, decompose the matrix

\[[m]=\left[\begin{array}{lll}4 & 2 & 1 \\2 & 6 & 2 \\1 & 2 & 8\end{array}\right]\]

in the form \([L][L]^{T}\).

Transcribed Image Text:

[m] = [L][L]T (E.1)