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Linear Algebra With Applications

Otto Bretscher

$Math Studyset Vaia Explanations$ Math

5th Edition · 442 pages

ISBN: 9780321796974

Chapter 1: Q25E (page 1)

If the positive definite matrix Ais similar to the symmetric matrix B, then Bmust be positive definite as well.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

It is given that A is a positive definite matrix. So,A is similar to a diagonal matrix whose all diagonal entries are positive.

Also,B is similar matrix to A.

So, there exists a non-singular matrix C such that BC=A.

As a result,BC is similar to a diagonal matrix whose all diagonal entries are positive.

Since C is non-singular, therefore is similar to a diagonal matrix whose all diagonal entries are positive.

Thus,B is also positive definite.

02

Final Answer

The given statement is TRUE.

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Most popular questions from this chapter

Consider the accompanying table. For some linear systems $A = \vec{x} = \vec{b}$ , you are given either the rank of the coefficient matrix $A$ , or the rank of the augmented matrix $[A : \vec{b}]$ . In each case, state whether the system could have no solution, one solution, or infinitely many solutions. There may be more than one possibility for some systems. Justify your answers.

Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This simple ride will not make any left or right turns; that is, the track lies in a vertical plane. The accompanying figure shows the ride as viewed from the side. The points $(a_{i}, b_{j})$ are given to you, and your job is to connect the dots in a reasonably smooth way. Let $a_{i + 1} > a_{i} for i = 0, . . . . . ., n - 1$ .

One method often employed in such design problems is the technique of cubic splines. We choose $f_{i} (t)$ , a polynomial of degree $\leq 3$ , to define the shape of the ride between $(a_{i - 1}, b_{i - 1})$ and $(a_{i}, b_{j}), for i = 0, . . . . ., n$ .

Obviously, it is required that $f_{i} (a_{i}) = b_{i}$ and $f_{i} (a_{i - 1}) = b_{i - 1}, for i = 0, . . . . . . ., n$ . To guarantee a smooth ride at the points $(a_{i}, b_{i})$ , we want the first and second derivatives of $f_{i}$ and $f_{i + 1}$ to agree at these points:

$f'_{i} (a_{i}) = f'_{i + 1} (a_{i})$ and

$f''_{i} (a_{i}) = f''_{i + 1} (a_{i}), for i = 0, . . . . . . ., n - 1 .$

Explain the practical significance of these conditions. Explain why, for the convenience of the riders, it is also required that

$f'_{1} (a_{0}) = f'_{n} (a_{n}) = 0$

Show that satisfying all these conditions amounts to solving a system of linear equations. How many variables are in this system? How many equations? (Note: It can be shown that this system has a unique solution.)

In Exercises 1 through 12, find all solutions of the equations

with paper and pencil using Gauss–Jordan elimination.

Show all your work.

$| \begin{matrix} x_{4} + 2 x_{5} - x_{6} = 2 \\ x_{1} + 2 x_{2} + x_{5} - x_{6} = 0 \\ x_{1} + 2 x_{2} + 2 x_{3 -} x_{5} - x_{6} = 2 \end{matrix} |$

a. Write the system in vector form.

$| \begin{matrix} x + 2 y = 7 \\ 3 x + y = 11 \end{matrix} |$

b. Use your answer in part (a) to represent the system geometrically. Solve the system and represent the solution geometrically.

Compute the products $A \vec{x}$ in Exercises 16 through 19 using paper and pencil (if the products are defined).

18. $[\begin{matrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{matrix}] [\begin{matrix} 1 \\ 2 \end{matrix}]$

See all solutions

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