Could you please help with question 309 and 310? Thanks

Exercise 309 Exercise 310 Theorem 3 Exercise 311 Exercise 312 75 5.8 Real symmetric matrices Real symmetric matrices play the role of real numbers in matrix analysis. Let A = A3 + LA; denote the real and imaginary parts of the m X to. matrix A. Show that _ A3 A; T(A)_(AI AR): is a faithful representation of the complex matrix A as a real matrix of twice the size, in the sense that for all complex matrices A and B o T{ocA) = ocT (A) . T{AH) = T(A)T o T{A + B) = T(A) + T(B) o T{AB) = T(A)T[B) whenever the operations are well-dened. Show that o T{unitary) = orthogonal o T(Hermitian) = symmetric o T(skewHermitian) = skew-symmetric Let A be a real symmetric matrix. Then there exists a real orthogonal matrix Q and a real diagonal matrix A such that A = QAQT and A\" Z Ai+l,i+1- Proof. Just repeat the proof of the Schur decomposition and observe that you can use orthogonal transforms instead of unitary transforms since the eigenvalues are known to be real. Also, symmetry will help to directly produce a diagonal rather than upper-triangular matrix. Work out a detailed proof. El From now on we will use the notation A, = A1- for convenience. Let A be a real m x to. matrix. Show that Az A ||A||2= max " "2: ax ' '42. OazEC" IIZII2 095x611?\" |X||2