2. Consider a fixed matrix M R2x2, and define the Lyapunov operator LM (Rx2, R)...

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2. Consider a fixed matrix M € R2x2, and define the Lyapunov operator LM (R²x2, R) → (R²x2, R) by LM(Q)=MTQ+QM, QER²X2 (a) Show that LM is a linear operator. -61 find A, the representation of LM with respect to the basis [9], (b) For the special case M = €1 = ez = =89₁ €3 = ē₁ = [], ²₂2=6 1], ²=[%], 1 e4 = (c) Compute the change of basis matrix mapping the representation of a vector with respect to the basis {e₁..4} to its representation with respect to the basis {₁4}, defined as 69 ē4 = -681₁ (d) Use MATLAB to compute A = PAP-1, the representation of the Lyapunov operator with respect to the basis {e₁..4}. 1Revised January 23, 2023. (e) With M as given in (2b), find bases for the range and nullspace of LM. (Recall that these subspaces lie in R²x2.) 2. Consider a fixed matrix M € R2x2, and define the Lyapunov operator LM (R²x2, R) → (R²x2, R) by LM(Q)=MTQ+QM, QER²X2 (a) Show that LM is a linear operator. -61 find A, the representation of LM with respect to the basis [9], (b) For the special case M = €1 = ez = =89₁ €3 = ē₁ = [], ²₂2=6 1], ²=[%], 1 e4 = (c) Compute the change of basis matrix mapping the representation of a vector with respect to the basis {e₁..4} to its representation with respect to the basis {₁4}, defined as 69 ē4 = -681₁ (d) Use MATLAB to compute A = PAP-1, the representation of the Lyapunov operator with respect to the basis {e₁..4}. 1Revised January 23, 2023. (e) With M as given in (2b), find bases for the range and nullspace of LM. (Recall that these subspaces lie in R²x2.)

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The image contains a set of multiple parts of a problem a to e based on linear algebra specifically relating to a Lyapunov operator change of basis an... View the full answer