91Ó°ÊÓ

Let \(A\) be a \(2 \times 2\) matrix and let \(p(\lambda)=\lambda^{2}+b \lambda+c\) be the characteristic polynomial of \(A .\) Show that \(b=-\operatorname{tr}(A)\) and \(c=\operatorname{det}(A)\)

Let \(A\) be an \(n \times n\) symmetric negative definite matrix. (a) What will the sign of \(\operatorname{det}(A)\) be if \(n\) is even? If \(n\) is odd? (b) Show that the leading principal submatrices of \(A\) are negative definite. (c) Show that the determinants of the leading principal submatrices of \(A\) alternate in sign.

Let \(A\) be an \(n \times n\) matrix and let \(\lambda\) be an eigenvalue of \(A .\) If \(A-\lambda I\) has rank \(k,\) what is the dimension of the eigenspace corresponding to \(\lambda ?\) Explain.

Let \(A\) be a diagonalizable \(n \times n\) matrix. Prove that if \(B\) is any matrix that is similar to \(A,\) then \(B\) is diagonalizable.

Let \(A\) be a \(n \times n\) matrix with Schur decomposition \(U T U^{H} .\) Show that if the diagonal entries of \(T\) are all distinct, then there is an upper triangular matrix \(R\) such that \(X=U R\) diagonalizes \(A\)

Let \(T\) be an upper triangular matrix with distinct diagonal entries (i.e., \(t_{i i} \neq t_{j j}\) whenever \(i \neq j\) ). Show that there is an upper triangular matrix \(R\) that diagonalizes \(T\)

Let \(Q\) be an orthogonal matrix. (a) Show that if \(\lambda\) is an eigenvalue of \(Q,\) then \(|\lambda|=1\) (b) Show that \(|\operatorname{det}(Q)|=1\)

Show that if \(A\) is skew Hermitian and \(\lambda\) is an eigenvalue of \(A\) then \(\lambda\) is purely imaginary (i.e., \(\lambda=b i\) where \(b\) is real).

Let \(Q\) be an orthogonal matrix with an eigenvalue \(\lambda_{1}=1\) and let \(\mathbf{x}\) be an eigenvector belonging to \(\lambda_{1}\) Show that \(\mathbf{x}\) is also an eigenvector of \(Q^{T}\)

Show that if two \(n \times n\) matrices \(A\) and \(B\) have a common eigenvector \(\mathbf{x}\) (but not necessarily a common eigenvalue), then \(\mathbf{x}\) will also be an eigenvector of any matrix of the form \(C=\alpha A+\beta B\)