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Chapter 3: Problem 2

A symmetric matrix is one for which the transpose of the matrix is the same as the original matrix, \(A^{T}=A\). An antisymmetric matrix is one that satisfies \(A^{T}=-A\). a. Show that the diagonal elements of an \(n \times n\) antisymmetric matrix are all zero. b. Show that a general \(3 \times 3\) antisymmetric matrix has three independent off-diagonal elements. c. How many independent elements does a general \(3 \times 3\) symmetric matrix have? d. How many independent elements does a general \(n \times n\) symmetric matrix have? e. How many independent elements does a general \(n \times n\) antisymmetric matrix have?

Short Answer

Expert verified
a. All diagonal elements of an antisymmetric matrix are zero. b. A 3x3 antisymmetric matrix has 3 independent off-diagonal elements. c. A 3x3 symmetric matrix has 6 independent elements. d. An nxn symmetric matrix has \(n(n+1)/2\) independent elements. e. An nxn antisymmetric matrix has \(n(n-1)/2\) independent elements.

Step by step solution

01

Diagonal Elements of Antisymmetric Matrix

The task is to show that the diagonal elements of an \(n \times n\) antisymmetric matrix are all zero. For any matrix A, the element at the intersection of the i-th row and the j-th column is denoted by \(A_{ij}\). Now, for an antisymmetric matrix, by definition, \(A^{T} = -A\). Hence, \(A_{ij} = -A_{ji}\). For diagonal elements, i = j, so \(A_{ii} = -A_{ii}\). The only number that equals its own negative is zero. So, all diagonal elements of an antisymmetric matrix are zero.
02

Independent Off-diagonal Elements of 3x3 Antisymmetric Matrix

To show that a general 3x3 antisymmetric matrix has 3 independent off-diagonal elements, it's necessary to construct the general form of such a matrix. A 3x3 antisymmetric matrix A would look like: \[\[0, a, b\], [-a, 0, c\], [-b, -c, 0]\]. Here, 'a', 'b', and 'c' are the independent non-diagonal elements. As it can be seen, there are 3 independent off-diagonal elements.
03

Independent Elements of 3x3 Symmetric Matrix

Symmetric matrices have the property that \(A_{ij}=A_{ji}\). Therefore, in a 3x3 symmetric matrix, the elements below the diagonal are the same as the elements above the diagonal. Hence, there are 3 diagonal elements and 3 independent pairs of off-diagonal elements, leading to a total of 6 independent elements in a 3x3 symmetric matrix.
04

Independent Elements of nxn Symmetric Matrix

In an \(n \times n\) symmetric matrix, each row (or column) has i independent elements, where i is the row (or column) number. The total number of independent elements would be the sum of independent elements in each row/column, which would sum up to \(n(n+1)/2\).
05

Independent Elements of nxn Antisymmetric Matrix

Similar to the symmetric matrix, an \(n \times n\) antisymmetric matrix has elements below the diagonal are negative of elements above the diagonal. However, unlike the symmetric matrix, diagonal elements must be zero. So, there are \(n(n-1)/2\) independent off-diagonal elements, as only half of the off-diagonal elements are distinct.

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Most popular questions from this chapter

Consider the matrix representations for two-dimensional rotations of vectors by angles \(\alpha\) and \(\beta\), denoted by \(R_{a}\) and \(R_{\beta}\), respectively. a. Find \(R_{a}^{-1}\) and \(R_{\alpha}^{T}\). How do they relate? b. Prove that \(R_{\alpha+\beta}=R_{\alpha} R_{\beta}=R_{\beta} R_{\alpha}\).

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