Problem 7 (Diagonalization). Let Symm := {A € Mn×n | A² = A} be the set of all n×n symmetric matrices with real coefficients and let Skewn {A Є Mnxn | At -A} be the set of all n×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 and Skew are subspaces of Mnxn. = n - Symm ☺ Skewn. Hint: note that A = ¾½ (A + A²) + ½ (A — A²). (b) Prove that Mnxn (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 Symm := {A € Mn×n | A² = A} be the set of all n×n symmetric matrices with real coefficients and let Skewn {A Є Mnxn | At -A} be the set of all n×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 and Skew are subspaces of Mnxn. = n - Symm ☺ Skewn. Hint: note that A = ¾½ (A + A²) + ½ (A — A²). (b) Prove that Mnxn (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
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section: Chapter Questions
Problem 14RQ
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Transcribed Image Text:Problem 7 (Diagonalization). Let Symm := {A € Mn×n | A² = A} be the set of all n×n symmetric matrices
with real coefficients and let Skewn {A Є Mnxn | At -A} be the set of all n×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 and Skew are subspaces of Mnxn.
=
n
- Symm ☺ Skewn. Hint: note that A = ¾½ (A + A²) + ½ (A — A²).
(b) Prove that Mnxn
(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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