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Problem 7 (Diagonalization). Let Symm₁ = {A = Mnxn | At = A} be the set of all nxn symmetric matrices with real coefficients and let Skewn := {A Є Mnxn | At = -A} be the set of all n xn 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, and Skew, are subspaces of Mnxn. (b) Prove that Mnxn = = Symm Skewn. Hint: note that A = ½½ (A + A²) + ½ (A — A²). (c) Define the function L: Mnxn → Mnxn by (i) Prove that L is a linear transformation. (ii) Prove that 0 and 2 are eigenvalues of L. (iii) Prove that L is diagonalizable. L(A) = A- At.

Problem 7 (Diagonalization). Let Symm₁ = {A = Mnxn | At = A} be the set of all nxn symmetric matrices with real coefficients and let Skewn := {A Є Mnxn | At = -A} be the set of all n xn 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, and Skew, are subspaces of Mnxn. (b) Prove that Mnxn = = Symm Skewn. Hint: note that A = ½½ (A + A²) + ½ (A — A²). (c) Define the function L: Mnxn → Mnxn by (i) Prove that L is a linear transformation. (ii) Prove that 0 and 2 are eigenvalues of L. (iii) Prove that L is diagonalizable. L(A) = A- At.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 3AEXP
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Problem 7 (Diagonalization). Let Symm₁ = {A = Mnxn | At = A} be the set of all nxn symmetric matrices with real coefficients and let Skewn := {A Є Mnxn | At = -A} be the set of all n xn 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, and Skew, are subspaces of Mnxn. (b) Prove that Mnxn = = Symm Skewn. Hint: note that A = ½½ (A + A²) + ½ (A — A²). (c) Define the function L: Mnxn → Mnxn by (i) Prove that L is a linear transformation. (ii) Prove that 0 and 2 are eigenvalues of L. (iii) Prove that L is diagonalizable. L(A) = A- At.
Transcribed Image Text:Problem 7 (Diagonalization). Let Symm₁ = {A = Mnxn | At = A} be the set of all nxn symmetric matrices with real coefficients and let Skewn := {A Є Mnxn | At = -A} be the set of all n xn 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, and Skew, are subspaces of Mnxn. (b) Prove that Mnxn = = Symm Skewn. Hint: note that A = ½½ (A + A²) + ½ (A — A²). (c) Define the function L: Mnxn → Mnxn by (i) Prove that L is a linear transformation. (ii) Prove that 0 and 2 are eigenvalues of L. (iii) Prove that L is diagonalizable. L(A) = A- At.
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