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Chapter 2: Problem 61

Prove that the main diagonal of a skew-symmetric matrix consists entirely of zeros.

Short Answer

Expert verified
The main diagonal of a skew-symmetric matrix consists entirely of zeros because each diagonal element, being equal to its own negation (due to the skew-symmetry), must be zero.

Step by step solution

01

Understanding the definition of skew-symmetric matrix

A skew-symmetric (or anti-symmetric) matrix is a square matrix whose transpose equals its negative. In other words, for a matrix A, if \( A^T = -A \), then A is a skew-symmetric matrix.
02

Selecting an element from the diagonal

Consider an arbitrary element \( a_{ii} \) from the main diagonal of the skew-symmetric matrix A. The main diagonal consists of all elements \( a_{ii} \) where i equals j.
03

Applying the property of skew-symmetry

Following the definition of skew-symmetry, the element in the i-th row and i-th column of matrix A must be equal to the negation of the element in the i-th row and i-th column of \( -A \), that is, \( a_{ii} = -(a_{ii}) \). The only number that is equal to its own negation is zero, so \( a_{ii} = 0 \). This argument applies to any element their chosen from the main diagonal.
04

Conclusion

Since the arbitrary element \( a_{ii} \) taken from the main diagonal of skew-symmetric matrix A must be zero, it follows that all elements of the main diagonal must be zero.

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Most popular questions from this chapter

Let \(P\) be a \(2 \times 2\) stochastic matrix. Prove that there exists a \(2 \times 1\) state matrix \(X\) with nonnegative entries such that \(P X=X\)

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The identity matrix is an elementary matrix. (b) If \(E\) is an elementary matrix, then \(2 E\) is an elementary matrix. (c) The matrix \(A\) is row-equivalent to the matrix \(B\) if there exists a finite number of elementary matrices \(E_{1}, E_{2}, \ldots, E_{k}\) such that \(A=E_{k} E_{k-1} \cdot \cdot \cdot E_{2} E_{1} B\) (d) The inverse of an elementary matrix is an elementary matrix.

Use elementary matrices to find the inverse of $$A=\left[\begin{array}{lll} 1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{lll} 1 & 0 & 0 \\ b & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & c \end{array}\right], c \neq 0$$

determine whether the matrix is stochastic. $$\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$

Factor the matrix \(A\) into a product of elementary matrices. $$A=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$$
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