Question

Ne discussed in class the relation between the eigenvalues of \( M \) and those of \( \left(\begin{array}{cc}0 & M \\ M & 0\end{array}\right) \). We saw in Assignment 2 the relation between the eigenv

Answer 1

The eigenvalues of
`\( \left(\begin{array}{cc}0 & M \\ M & 0\end{array}\right) \)`are the negative and positive counterparts of the eigenvalues of matrix ( M ).

Explanation:

The eigenvalues of ( M ) are mirrored in the eigenvalues of \
`(\left(\begin{array}{cc}0 & M \\ M & 0\end{array}\right) \),`resulting in pairs of positive and negative values. This phenomenon occurs due to the structure of the block matrix, where swapping the blocks diagonally yields the new matrix. The characteristic equation of
`\( \left(\begin{array}{cc}0 & M \\ M & 0\end{array}\right) \)`involves determinants of the form
`\( \text{det}(M - \lambda I) \cdot \text{det}(-M - \lambda I) \),`producing eigenvalues that are negatives of each other. When \( M \) has eigenvalues
`\( \lambda \),` then the eigenvalues of
`\( \left(\begin{array}{cc}0 & M \\ M & 0\end{array}\right) \) are \( \lambda \) and \( -\lambda \)`. This relationship demonstrates a direct correspondence between the eigenvalues of \( M \) and those of the block matrix.

The eigenvalues of
`\( \left(\begin{array}{cc}0 & M \\ M & 0\end{array}\right) \)` exhibit a distinct pattern in relation to those of matrix ( M ). Understanding this pattern involves observing the structure of the block matrix. When examining the characteristic equation of the block matrix, the determinant comprises terms involving
`\( \text{det}(M - \lambda I) \) and \( \text{det}(-M - \lambda I) \).` Consequently, the eigenvalues of
`\( \left(\begin{array}{cc}0 & M \\ M & 0\end{array}\right) \)`manifest as pairs: for every eigenvalue
`\( \lambda \) of \( M \),` the block matrix yields eigenvalues
`\( \lambda \) and \( -\lambda \).` This duality reflects a clear relationship between the eigenvalues of ( M ) and those of the constructed block matrix, showcasing a symmetric interplay between positive and negative values within their respective spectra.