|(1) Show that if A is an n × n symmetric, positive semi-definite matrix, then A must have nonnega- tive diagonal elements. |(ii) Show that if A is an n × n symmetric, positive definite matrix, then A must have strictly positive diagonal elements. (iii) Write down a 2 × 2 symmetric matrix with strictly positive diagonal elements that is not positive definite.

|(1) Show that if A is an n × n symmetric, positive semi-definite matrix, then A must have nonnega- tive diagonal elements. |(ii) Show that if A is an n × n symmetric, positive definite matrix, then A must have strictly positive diagonal elements. (iii) Write down a 2 × 2 symmetric matrix with strictly positive diagonal elements that is not positive definite.

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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In general, it is difficult to show that two matrices are similar. However, if two similar matrices are diagonalizable, the task becomes easier. In Exercises 38-41, show that A and B are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that .
Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a 22 matrix.
11. Find two nonzero matrices and such that.

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