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Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![Consider the following 2 x 2 matrix
A =
For any matrix, we have a clear understanding of some exponents: A" refers to multiplying a matrix to
itself n times and A-l refers to the inverse matrix. Combining this idea gives A-n = (A-!)", the nth
power of the inverse.
A natural question to ask is, can we take the nth root of a matrix? We're going to try to find A/3, i.e. a
matrix that when we cube it gives A.
(a) Calculate the eigenvalues and eigenvectors of A.
(b) Find a matrix P and a diagonal matrix D such that A = PDP-1.
(c) Find a matrix E such that E = D. Explain why this result is simple compared to finding the cube
root of A.
(d) Using part (c), find a matrix B such that B3 = A.](https://content.bartleby.com/qna-images/question/dcf49873-c5b0-4558-94e1-142bcbf716cd/40198c9a-f9ce-4f6a-8512-c4b97dee3df0/9wuwbqs_thumbnail.jpeg)
Transcribed Image Text:Consider the following 2 x 2 matrix
A =
For any matrix, we have a clear understanding of some exponents: A" refers to multiplying a matrix to
itself n times and A-l refers to the inverse matrix. Combining this idea gives A-n = (A-!)", the nth
power of the inverse.
A natural question to ask is, can we take the nth root of a matrix? We're going to try to find A/3, i.e. a
matrix that when we cube it gives A.
(a) Calculate the eigenvalues and eigenvectors of A.
(b) Find a matrix P and a diagonal matrix D such that A = PDP-1.
(c) Find a matrix E such that E = D. Explain why this result is simple compared to finding the cube
root of A.
(d) Using part (c), find a matrix B such that B3 = A.
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