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Chapter 8: Q20E (page 413)
Matrix is diagonalizable.
The given statement is TRUE.
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Consider a symmetric matrixA. If the vector is in the image of Aand is in the kernel of A, is necessarily orthogonal to? Justify your answer.
A Cholesky factorization of a symmetric matrix A is a factorization of the formwhere L is lower triangular with positive diagonal entries.
Show that for a symmetricmatrix A, the following are equivalent:
(i) A is positive definite.
(ii) All principal submatricesrole="math" localid="1659673584599" of A are positive definite. See
Theorem 8.2.5.
(iii)
(iv) A has a Cholesky factorization
Hint: Show that (i) implies (ii), (ii) implies (iii), (iii) implies (iv), and (iv) implies (i). The hardest step is the implication from (iii) to (iv): Arguing by induction on n, you may assume that ) has a Cholesky factorization . Now show that there exist a vector and a scalar t such that
Explain why the scalar t is positive. Therefore, we have the Cholesky factorization
This reasoning also shows that the Cholesky factorization of A is unique. Alternatively, you can use the LDLT factorization of A to show that (iii) implies (iv).See Exercise 5.3.63.
To show that (i) implies (ii), consider a nonzero vector, and define
role="math" localid="1659674275565"
In (fill in n 鈭� m zeros). Then
role="math" localid="1659674437541"
Determine the definiteness of the quadratic forms in Exercises 4 through 7.
4.
Find a symmetric 2x2matrix Bsuch that
Consider a symmetric nxnmatrix A with. Is the linear transformationnecessarily the orthogonal projection onto a subspace of?
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