Exercises 5C The row echelon form of the matrix of an operator does not give us a list of the eigenvalues of the operator. In contrast, an upper-triangular matrix with respect to some basis gives us a list of all the eigenvalues of the op- erator. However, there is no method for computing exactly such an upper- triangular matrix, even though 5.47 guarantees its existence if 𝐅 = 𝐂. [[5C-1]] Prove or give a counterexample: If 𝑇 ∈ L(𝑉) and 𝑇2 has an upper-triangular matrix with respect to some basis of 𝑉, then 𝑇 has an upper-triangular matrix with respect to some basis of 𝑉. Section 5C Upper-Triangular Matrices 161 [[5C-2]]. 2  Suppose 𝐴 and 𝐵 are upper-triangular matrices of the same size, with 𝛼1, ..., 𝛼𝑛 on the diagonal of 𝐴 and 𝛽1, ..., 𝛽𝑛 on the diagonal of 𝐵. 1. (a)  Show that 𝐴 + 𝐵 is an upper-triangular matrix with 𝛼1+𝛽1,...,𝛼𝑛+𝛽𝑛 on the diagonal. 2. (b)  Show that 𝐴𝐵 is an upper-triangular matrix with 𝛼1𝛽1, ..., 𝛼𝑛𝛽𝑛 on the diagonal. The results in this exercise are used in the proof of 5.81. [[5C-3]]. 3  Suppose 𝑇 ∈ L(𝑉) is invertible and 𝑣1, ..., 𝑣𝑛 is a basis of 𝑉 with respect to which the matrix of 𝑇 is upper triangular, with 𝜆1, ..., 𝜆𝑛 on the diagonal. Show that the matrix of 𝑇−1 is also upper triangular with respect to the basis 𝑣1, ..., 𝑣𝑛, with 𝜆1 , ... , 𝜆1 1𝑛 on the diagonal. [[5C-4]]. 4  Give an example of an operator whose matrix with respect to some basis contains only 0’s on the diagonal, but the operator is invertible. This exercise and the exercise below show that 5.41 fails without the hypothesis that an upper-triangular matrix is under consideration. [[5C-5]]. 5  Give an example of an operator whose matrix with respect to some basis contains only nonzero numbers on the diagonal, but the operator is not invertible. [[5C-6]]. 6  Suppose 𝐅 = 𝐂, 𝑉 is finite-dimensional, and 𝑇 ∈ L(𝑉). Prove that if 𝑘 ∈ {1, ..., dim 𝑉}, then 𝑉 has a 𝑘-dimensional subspace invariant under 𝑇. [[5C-7]]. 7  Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and 𝑣 ∈ 𝑉. 1. (a)  Prove that there exists a unique monic polynomial 𝑝𝑣 of smallest degree such that 𝑝𝑣(𝑇)𝑣 = 0. 2. (b)  Prove that the minimal polynomial of 𝑇 is a polynomial multiple of 𝑝𝑣. [[5C-8]]. 8  Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and there exists a nonzero vector 𝑣 ∈ 𝑉 such that 𝑇2𝑣 + 2𝑇𝑣 = −2𝑣. 1. (a)  Prove that if 𝐅=𝐑, then there does not exist a basis of 𝑉 with respect to which 𝑇 has an upper-triangular matrix. 2. (b)  Prove that if 𝐅 = 𝐂 and 𝐴 is an upper-triangular matrix that equals the matrix of 𝑇 with respect to some basis of 𝑉, then −1 + 𝑖 or −1 − 𝑖 appears on the diagonal of 𝐴. [[5C-9]]. 9  Suppose 𝐵 is a square matrix with complex entries. Prove that there exists an invertible square matrix 𝐴 with complex entries such that 𝐴−1𝐵𝐴 is an upper-triangular matrix. [[5C-10]]. 10  Suppose𝑇∈L(𝑉)and𝑣1,...,𝑣𝑛isabasisof𝑉.Showthatthefollowing are equivalent. (a) The matrix of 𝑇 with respect to 𝑣1, ..., 𝑣𝑛 is lower triangular. (b) span(𝑣𝑘,...,𝑣𝑛)isinvariantunder𝑇foreach𝑘=1,...,𝑛. (c) 𝑇𝑣𝑘 ∈ span(𝑣𝑘,...,𝑣𝑛) for each 𝑘 = 1,...,𝑛. A square matrix is called lower triangular if all entries above the diagonal are 0. [[5C-11]]. 11  Suppose 𝐅 = 𝐂 and 𝑉 is finite-dimensional. Prove that if 𝑇 ∈ L(𝑉), then there exists a basis of 𝑉 with respect to which 𝑇 has a lower-triangular matrix. [[5C-12]]. 12  Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉) has an upper-triangular matrix with respect to some basis of 𝑉, and 𝑈 is a subspace of 𝑉 that is invariant under 𝑇. (a) Provethat𝑇|𝑈hasanupper-triangularmatrixwithrespecttosomebasis of 𝑈. (b) Provethatthequotientoperator𝑇/𝑈hasanupper-triangularmatrixwith respect to some basis of 𝑉/𝑈. The quotient operator 𝑇/𝑈 was defined in Exercise 38 in Section 5A. [[5C-13]]. 13  Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Suppose there exists a subspace 𝑈 of 𝑉 that is invariant under 𝑇 such that 𝑇|𝑈 has an upper- triangular matrix with respect to some basis of 𝑈 and also 𝑇/𝑈 has an upper-triangular matrix with respect to some basis of 𝑉/𝑈. Prove that 𝑇 has an upper-triangular matrix with respect to some basis of 𝑉. [[5C-14]]. 14  Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Prove that 𝑇 has an upper- triangular matrix with respect to some basis of 𝑉 if and only if the dual operator 𝑇′ has an upper-triangular matrix with respect to some basis of the dual space 𝑉′.