Problem 7 (Diagonalization). Let Symm with real coefficients and let Skewn: == === {A Є Mnxn | At = A} be the set of all nxn symmetric matrices {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. (b) Prove that Mnxn = Symm„ ☺ Skewɲ. Hint: note that A = ½ (A+ A²) + ½ (A - At). (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 with real coefficients and let Skewn: == === {A Є Mnxn | At = A} be the set of all nxn symmetric matrices {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. (b) Prove that Mnxn = Symm„ ☺ Skewɲ. Hint: note that A = ½ (A+ A²) + ½ (A - At). (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. :=
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.3: Matrix Algebra
Problem 85E: Determine if the statement is true or false. If the statement is false, then correct it and make it...
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Transcribed Image Text:Problem 7 (Diagonalization). Let Symm
with real coefficients and let Skewn:
==
===
{A Є Mnxn | At = A} be the set of all nxn symmetric matrices
{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.
(b) Prove that Mnxn = Symm„ ☺ Skewɲ. Hint: note that A = ½ (A+ A²) + ½ (A - At).
(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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