Let A^T=-A be a real, skew-symmetric n ×n matrix. (a) Prove that the only possible real eigenval

step 1 we are given a matrix A, which is a real N by N -by -end -scequimetric matrix, that N is some od

Let A^T=-A be a real, skew-symmetric n ×n matrix. (a) Prove...

Let $A^T=-A$ be a real, skew-symmetric $n \times n$ matrix. (a) Prove that the only possible real eigenvalue of $A$ is $\lambda=0$. (b) More generally, prove that all eigenvalues $\lambda$ of $A$ are purely imaginary, i.e., $\operatorname{Re} \lambda=0$. (c) Explain why 0 is an eigenvalue of $A$ whenever $n$ is odd. (d) Explain why, if $n=3$, the eigenvalues of $A \neq 0$ are $0, i \omega,-i \omega$, for some real $\omega \neq 0$. (e) Verify these facts for the particular matrices (i) $\left(\begin{array}{rr}0 & -2 \\ 2 & 0\end{array}\right)$, (ii) $\left(\begin{array}{rrr}0 & 3 & 0 \\ -3 & 0 & -4 \\ 0 & 4 & 0\end{array}\right)$, (iii) $\left(\begin{array}{rrr}0 & 1 & -1 \\ -1 & 0 & -1 \\ 1 & 1 & 0\end{array}\right)$, (kv) $\left(\begin{array}{rrrr}0 & 0 & 2 & 0 \\ 0 & 0 & 0 & -3 \\ -2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0\end{array}\right)$.

Let $A^T=-A$ be a real, skew-symmetric $n \times n$ matrix. (a) Prove that the only possible real eigenvalue of $A$ is $\lambda=0$. (b) More generally, prove that all eigenvalues $\lambda$ of $A$ are purely imaginary, i.e., $\operatorname{Re} \lambda=0$. (c) Explain why 0 is an eigenvalue of $A$ whenever $n$ is odd. (d) Explain why, if $n=3$, the eigenvalues of $A \neq 0$ are $0, i \omega,-i \omega$, for some real $\omega \neq 0$. (e) Verify these facts for the particular matrices
(i) $\left(\begin{array}{rr}0 & -2 \\ 2 & 0\end{array}\right)$,
(ii) $\left(\begin{array}{rrr}0 & 3 & 0 \\ -3 & 0 & -4 \\ 0 & 4 & 0\end{array}\right)$,
(iii) $\left(\begin{array}{rrr}0 & 1 & -1 \\ -1 & 0 & -1 \\ 1 & 1 & 0\end{array}\right)$,
(kv) $\left(\begin{array}{rrrr}0 & 0 & 2 & 0 \\ 0 & 0 & 0 & -3 \\ -2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0\end{array}\right)$.

Key Concepts

Eigenvalue Structure in Low Dimensions

For low-dimensional cases, such as 3 × 3 matrices, the spectrum of a non-zero skew?symmetric matrix typically consists of one zero eigenvalue and a pair of purely imaginary eigenvalues, often expressed as i? and -i? for some non-zero real number ?. This specific structure illustrates the general theoretical results and provides a tangible example of how these abstract concepts manifest in smaller matrices.

Implications of Matrix Dimension (Odd Dimension)

An important consequence of these spectral properties is that if the matrix size is odd, the characteristic polynomial, being of odd degree, must have at least one real eigenvalue. Given that the only allowable real eigenvalue for a skew?symmetric matrix is zero, it follows that zero must appear in the spectrum. This reasoning is based on general properties of polynomials and the structure imposed by the skew?symmetric condition.

Skew?Symmetric Matrices

A skew?symmetric matrix is one that satisfies the condition A^T = -A. This property has far-reaching implications, such as the fact that its quadratic form v^T A v always vanishes for any real vector v, and that its structure ensures certain symmetries in its spectrum. Understanding skew?symmetric matrices is crucial in many applications including differential equations and physics, where they often represent rotations or angular momentum operators.

Eigenvalue Properties of Real Matrices

For any real matrix, including skew?symmetric matrices, the eigenvalues typically come in complex conjugate pairs because the coefficients of the characteristic polynomial are real. This concept reinforces why non-real eigenvalues must be paired with their complex conjugates, and it provides the necessary algebraic foundation for understanding the spectrum of the matrix.

Purely Imaginary Eigenvalues

In the context of skew?symmetric matrices, the property A^T = -A forces any nonzero eigenvalue to be purely imaginary. This follows from observing that if v is a real eigenvector associated with a real eigenvalue, the quadratic form contradiction implies the eigenvalue must equal zero, and more generally, the inner product relations necessitate that any non-real eigenvalues must have zero real part.

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