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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q42E (page 291)

Consider an $n 脳 m$ matrix $A = Q R,$ where $Q$ is a $n 脳 m$ matrix with Orthonormality columns and $R$ is an upper triangular $m 脳 m$ matrix with positive diagonal entries $r_{11}, . . ., r_{m m}$ . Express $d e t (A^{T} A)$ in terms of the scalars $r_{i i}$ . What can you say about the sign of $d e t (A^{T} A)$ ?

Short Answer

Expert verified

Therefore, the $\det (A^{T} A)$ sign is always positive.

Step by step solution

01

Step by Step Solution: Step 1: Matrix Definition

Matrixis aset of numbers arranged inrowsandcolumnsso as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are $m$ rows and $n$ columns, the matrix is said to be a 鈥� $m$ by $n$ 鈥� matrix, written 鈥� $m 脳 n$ .鈥�

02

Step 2: To find det(ATA) is positive or negative

To solve,

$d e t (A^{T} A) = d e t ({(Q R)}^{T} Q R) d e t (A^{T} A) = d e t (R^{T} Q^{T} Q R) d e t (A^{T} A) = d e t (R^{T} R) d e t (A^{T} A) = d e t R^{T} d e t R d e t (A^{T} A) = d e t R d e t R d e t (A^{T} A) = (d e t R)^{2}$

$\det (A^{T} A) = {({鈭�}_{i = 1}^{m} r_{i, i})}^{2} > 0$

Therefore, the $\det (A^{T} A)$ sign is always positive.

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Most popular questions from this chapter

$d e t (4 A) = 4 d e t A$ for all $4 脳 4$ matricesA.

(For those who have studied multivariable calculus.) Let Tbe an invertible linear transformation from ${鈩�}^{2}$ to ${鈩�}^{2}$ , represented by the matrix M. Let $惟_{1}$ be the unit square in ${鈩�}^{2}$ and $惟_{2}$ its image under T . Consider a continuous function $f (x, y)$ from ${鈩�}^{2}$ to $鈩�$ , and define the function $g (u, v) = f (T (u, v))$ . What is the relationship between the following two double integrals?

${鈭�}_{惟 2} f (x, y) dA$ and ${鈭�}_{惟 1} g (u, v) dA$

Your answer will involve the matrix M. Hint: What happens when $f (x, y) = 1$ , for all $x, y$ ?

Vandermonde determinants (introduced by Alexandre-Th茅ophile Vandermonde). Consider distinct real numbers $a_{0}, a_{1}, . . . . ., a_{n .}$ . We define $(n + 1) 脳 (n + 1)$ the matrix

$A = [\begin{matrix} 1 & 1 & . . . . 1 \\ a_{0} & a_{1} & . . . . a_{n} \\ a_{0}^{2} & a_{1}^{2} & . . . . a_{1}^{2} \\ a_{0}^{n} & a_{1}^{n} & . . . . a_{n}^{n} \end{matrix}]$

Vandermonde showed that

$d e t (A) = \underset{i > j}{鈭�} (a_{i} - a_{j})$

the product of all differences $(a_{i} - a_{j})$ , where exceeds j.
a. Verify this formula in the case of $n = 1$ .
b. Suppose the Vandermonde formula holds for $n = 1$ . You are asked to demonstrate it for n. Consider the function

$f (t) = \det [\begin{matrix} 1 & 1 & . . . & 1 & 1 \\ a_{0} & a_{1} & . . . & a_{n - 1} & t \\ a_{0}^{2} & a_{1}^{2} & . . . & a_{n - 1} & t^{2} \\ 鈰� & 鈰� & . . . & 鈰� & 鈰� \\ a_{0}^{n} & a_{1}^{n} & . . . & a_{n - 1}^{n} & t^{n} \end{matrix}]$

Explain why f(t) is a polynomial of $n^{t h}$ degree. Find the coefficient k of $t^{n}$ using Vandermonde's formula for $a_{0}, . . ., a_{n - 1}$ . Explain why

role="math" localid="1659522435181" $f (a_{0}) = f (a_{1}) = . . . = f (a_{n - 1}) = 0$

Conclude that

$f (t) = k (t - a_{0}) (t - a_{1}) . . . (t - a_{n - 1})$

for the scalar k you found above. Substitute $t = a_{n}$ to demonstrate Vandermonde's formula.

There exist invertible $2 脳 2$ matrices A andBsuch that $d e t (A + B) = d e t A + d e t B$ .

$d e t [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{matrix}] = 1$

See all solutions

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