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Let TEL(C7) be defined by T(21, 22, 23, 24, 25, 26, 27) = (1+z1+z2+23+24, 7z2+23+24, T23+24, 724, V725+26+27, V726+27, V727) Let B,(C7) = {(1, 62, 63, 64, 65, 66, 67} be the standard basis of C7 (a) (25 pts

#### Let TEL(C7) be defined by T(21, 22, 23, 24, 25, 26, 27) = (1+z1+z2+23+24, 7z2+23+24, T23+24, 724, V725+26+27, V726+27, V727) Let B,(C7) = {(1, 62, 63, 64, 65, 66, 67} be the standard basis of C7 (a) (25 pts

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Let TEL(C7) be defined by T(21, 22, 23, 24, 25, 26, 27) = (1+z1+z2+23+24, 7z2+23+24, T23+24, 724, V725+26+27, V726+27, V727) Let B,(C7) = {(1, 62, 63, 64, 65, 66, 67} be the standard basis of C7 (a) (25 pts.) Find M(T,B,C)) (b) (25 pts.) Find the eigenvalues {x}k=1,...,? (c) For each eigenvalue, lk: i. (30 pts.) Find the eigenspace E(Ak, T) ii. (30 pts.) Find the generalized eigenspace G(Xk, T) (d) (30 pts.) What does the Jordan Form of MT, B,(C7)) look like. (e) Identify the Jordan Basis, i.e. the basis with respect to which M(MT, B,(C7), B1) is in Jordan Form: i. (15 pts.) Identify the Jordan Chains ii. (15 pts.) Collect the Jordan Basis, BJ 3. Let TEL(V), and B be an orthonormal basis, so that 3 1 3 1 3 MT,B) = 3 1 3 (a) (5+20 pts) Is T self-adjoint? Why/Why Not? (b) (5+20 pts) Is T normal? Why/Why Not? (c) (90 pts 2+3 pts/box with explanation) Now, let RE L(V) be a self-adjoint operator, SeL(V) a normal operator, and U EL(V) an operator that is neither self-adjoint nor normal; what properties do these operators have CLEARLY indicate ONE of R (if true only for F = R) / C (if true only for F = C) / F (always true) / Ninever / S-O:sometimes-other): 1 op = R, S, or U R, Self Adjoint S, Normal U, Neither (Not Self-Adjoint): 2 3 @ 3 ortho-normal basis: Mop, B) is upper triangular RCF NS- ORCF NS-O RCF NS-O 2 6 3 ortho-normal basis: Mop, B) is diagonal RCF NS- ORCF NS-O RCF NS-O all eigenvalues of op are real RCF NS- ORCF NS-O RCF NS-O TO op has positive square root RCF NS- ORCF NS-O RCF NS-O op has polar decomposition RCF NS-O'RCF NS-O RCF NS-O © RCF NS-O op has singular value decomposition R C F NS-O RCF NS-O On the next page (not in the boxes above!!!) for each answer in the grid give a SHORT EXPLANATION (why/why not, or what is the "extra condition(s)" needed for the "sometimes-other" properties? Example: “Skew-Normal-Pseudo-Adjoint Operators over the Sedenions are guaranteed spatio-temporally meta-invariant Blomgren-values by the Lothbrok-Aethelwulf Theorem." 1) 3 4 6 6 7 ® 9 @ © a) © f ® NOTE: If you use a separate sheet of paper, give the answers in ORDER 1-9,a-i, and CLEARLY indicate what is what. Non-sequential answers -NO CREDIT