Q54E If A is a positive semidefinite ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q54E (page 402)

If A is a positive semidefinite matrix with $a_{11} = 0$ , what can you say about the other entries in the first row and in the first column of A? Hint: Exercise 53 is helpful.

Short Answer

Expert verified

All of 's remaining entries must be zero.

Step by step solution

Describe the other entries in the first row and column of the A matrix

A is a n-dimensional positive semi-definite matrix with $a_{11} = 0$ Let $q (x) = x^{T} Ax$ be a quadratic form defined by A, where A is a variable. For any two integers i and j such that $1 鈮� i 鈮� j 鈮� n$ , the function p defined as $p (x, y) = q (0, 0, . . ., x_{(i)}, 0, . . . ., y_{(j)}, 0, . . . 0)$ produces a quadratic form with

$p (x, y) = [x y] [\begin{matrix} a_{ii} & a_{ij} \\ a_{ji} & a_{jj} \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$

and is positive semi-definite when q is positive semi-definite. i = 1 and for j = 2, 3, ...,n, we have

$[\begin{matrix} a_{11} & a_{1 j} \\ a_{j 1} & a_{jj} \end{matrix}] = [\begin{matrix} 0 & a_{1 j} \\ a_{j 1} & a_{jj} \end{matrix}]$

Semi definite affirmative.

Is a semi-definite affirmative setting. To put it another way, we'll need.

$|\begin{matrix} 0 & a_{1 j} \\ a_{j 1} & a_{jj} \end{matrix}| 鈮� 0 鈬� a_{1 j} a_{j 1} 鈮� 0 鈬� a_{1 j}^{2} 鈮� 0 鈬� a_{1 j} = 0$

For each j = 2, 3,...,n . As a result, all of A's first row must be zero.

All of 's remaining entries must be zero.

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91影视

If A is a positive semidefinite matrix with $a_{11} = 0$ , what can you say about the other entries in the first row and in the first column of A? Hint: Exercise 53 is helpful.

Short Answer

Step by step solution

Describe the other entries in the first row and column of the A matrix

Semi definite affirmative.

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91影视

If A is a positive semidefinite matrix with a11=0, what can you say about the other entries in the first row and in the first column of A? Hint: Exercise 53 is helpful.

Short Answer

Step by step solution

Describe the other entries in the first row and column of the A matrix

Semi definite affirmative.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

If A is a positive semidefinite matrix with $a_{11} = 0$ , what can you say about the other entries in the first row and in the first column of A? Hint: Exercise 53 is helpful.