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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2021 • Question 3

Question

Math 6310 • Algebra Spring 2021 • Question 3

Let \(I_k\) denote the \(k\times k\) identity matrix. Suppose \(M\) is a \(2n\times 2n\) matrix over \(\mathbb {Q}\) such that \(M^2=-I_{2n}\), prove that \(M\) is similar to the matrix

\[ \begin {pmatrix} 0&-I_n\\ I_n&0 \end {pmatrix}. \]

Raw LaTeX

Let \(I_k\) denote the \(k\times k\) identity matrix. Suppose \(M\) is a \(2n\times 2n\) matrix over \(\mathbb{Q}\) such that \(M^2=-I_{2n}\), prove that \(M\)
is similar to the matrix

\[
\begin{pmatrix}
0&-I_n\\
I_n&0
\end{pmatrix}.
\]

Topic Tags

Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization