3.27 Diagonal elements of Cholesky factor. Each X € S has a unique Cholesky factorization X = LLT, where L is lower triangular, with Lii > 0. Show that Lii is a concave function of X (with domain S2+). Hint. Lii can be expressed as Lii = (w - zTY-¹2)¹/2, where Y is the leading ix i submatrix of X. Z W ]

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3.27 Diagonal elements of Cholesky factor. Each X E S has a unique Cholesky factorization
X = LLT, where L is lower triangular, with Lü > 0. Show that Lii is a concave function
of X (with domain S).
Hint. Lii can be expressed as Lii = (w – zTY-12)/2, where
Y
T
w
is the leading i xi submatrix of X.
Transcribed Image Text:3.27 Diagonal elements of Cholesky factor. Each X E S has a unique Cholesky factorization X = LLT, where L is lower triangular, with Lü > 0. Show that Lii is a concave function of X (with domain S). Hint. Lii can be expressed as Lii = (w – zTY-12)/2, where Y T w is the leading i xi submatrix of X.
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