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Chapter 5: Q22E (page 263)

If Ais any square matrix, then matrix 12(A-AT) is skew-symmetric.

Short Answer

Expert verified

True.

Step by step solution

01

Step by step solution Step 1: Skew-symmetric matrices

Consider an m脳nmatrix A.

The transposeATof Ais then脳mmatrix whoseijthentry isjiththe entry of A: The roles of rows and columns are reversed.

We say that a square matrix Ais symmetricif AT=A, and Ais called skew-symmetricif AT=A.

02

Determine whether the statement is true or false

Consider the equation below.

(1/2(A-AT))=1/2(AT-A)=-1/2(A-AT)

Thus, by the definition of skew symmetric matrices statement is to be true.

Hence, the statement is true.

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Most popular questions from this chapter

Question: Consider an n脳mmatrix A, a vector in Rm, and a vector w鈫�in Rm. Show that (Av鈫�).w鈫�=v鈫�.(ATw鈫�).

Leonardo da Vinci and the resolution of forces. Leonardo (1452鈥�1519) asked himself how the weight of a body, supported by two strings of different length, is apportioned between the two strings.

Three forces are acting at the point D: the tensions F1and F2in the strings and the weight W鈫�. Leonardo believed that

||F鈫�1||||F鈫�2||=E炉A炉E炉B炉

Was he right? (Source: Les Manuscripts de L茅onard de Vinci, published by Ravaisson-Mollien, Paris, 1890.)

Hint: ResolveF1into a horizontal and a vertical component; do the same for F2. Since the system is at rest, the equationF鈫�1+F鈫�2+W鈫�=0鈫�holds. Express the ratios

||F鈫�1||||F鈫�2||and E炉A炉E炉B炉. In terms of伪and 尾, using trigonometric functions, and compare the results.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

10.[1111],[6464]

TRUE OR FALSE?If x鈬赌andy鈬赌are two vectors inRn, then the equation role="math" localid="1659506190737" 鈭�x鈫�+y鈫�鈭�2=鈭�x鈫�鈭�2+鈭�y鈫�鈭�2must hold.

Let Abe the matrix of an orthogonal projection. FindA2 in two ways:

a.Geometrically. (Consider what happens when you apply an orthogonal projection twice.)

b.By computation, using the formula given in Theorem 5.3.10
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