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5. Definition 4: Matrix WE Rnxn is said to be normal if WTW = WWT. (a) Prove that if A is a symmetric or a skew-symmetric matrix, then A is also a normal matrix. In other words, sets of n x n symmetric and skew-symmetric matrices are subsets of the set of n x n normal matrices. (b) Find a matrix X € R²×2 that is normal, but not symmetric or skew-symmetric. Ideally, you will provide a family of such matrices which will depend on one or more parameters. (c) Let T E Rnxn be an upper triangular matrix that is also normal. Prove that T must be a diagonal matrix.

5. Definition 4: Matrix WE Rnxn is said to be normal if WTW = WWT. (a) Prove that if A is a symmetric or a skew-symmetric matrix, then A is also a normal matrix. In other words, sets of n x n symmetric and skew-symmetric matrices are subsets of the set of n x n normal matrices. (b) Find a matrix X € R²×2 that is normal, but not symmetric or skew-symmetric. Ideally, you will provide a family of such matrices which will depend on one or more parameters. (c) Let T E Rnxn be an upper triangular matrix that is also normal. Prove that T must be a diagonal matrix.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.4: Similarity And Diagonalization
Problem 41EQ: In general, it is difficult to show that two matrices are similar. However, if two similar matrices...
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5. Definition 4: Matrix W € Rn×n is said to be normal if WTW = WWT. (a) Prove that if A is a symmetric or a skew-symmetric matrix, then A is also a normal matrix. In other words, sets of n x n symmetric and skew-symmetric matrices are subsets of the set of n x n normal matrices. (b) Find a matrix X € R²×² that is normal, but not symmetric or skew-symmetric. Ideally, you will provide a family of such matrices which will depend on one or more parameters. (c) Let T E Rnxn be an upper triangular matrix that is also normal. Prove that I must be a diagonal matrix.
Transcribed Image Text:5. Definition 4: Matrix W € Rn×n is said to be normal if WTW = WWT. (a) Prove that if A is a symmetric or a skew-symmetric matrix, then A is also a normal matrix. In other words, sets of n x n symmetric and skew-symmetric matrices are subsets of the set of n x n normal matrices. (b) Find a matrix X € R²×² that is normal, but not symmetric or skew-symmetric. Ideally, you will provide a family of such matrices which will depend on one or more parameters. (c) Let T E Rnxn be an upper triangular matrix that is also normal. Prove that I must be a diagonal matrix.
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