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(Solved): Problem 7 (Diagonalization). Let Symm_(n):={A in M_(n\times n)|A^(t)=A} be the set of all n\times n ...
Problem 7 (Diagonalization). Let Symm_(n):={A in M_(n\times n)|A^(t)=A} be the set of all n\times n symmetric matrices
with real coefficients and let Skew_(n):={A in M_(n\times n)|A^(t)=-A} be the set of all n\times n skew-symmetric matrices
with real coefficients. For this problem, feel free to use any properties of the matrix transpose you might find
useful.
(a) Prove that Symm_(n) and Skew_(n) are subspaces of M_(n\times n).
(b) Prove that M_(n\times n)=Symm_(n)o+Skew_(n). Hint: note that A=(1)/(2)(A+A^(t))+(1)/(2)(A-A^(t)).
(c) Define the function L:M_(n\times n)->M_(n\times n) by
L(A):=A-A^(t).
(i) Prove that L is a linear transformation.
(ii) Prove that 0 and 2 are eigenvalues of L.
(iii) Prove that L is diagonalizable.