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 Post subject: Can the following matrix be diagonalized?Posted: Thu, 19 Apr 2012 00:22:25 UTC
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Joined: Wed, 28 Sep 2011 23:37:54 UTC
Posts: 23
Hello, I'm fairly new to matrix algebra, and I'm iffy on how to properly find diagonalizations, and determining whether a matrix can be diagonalized. I'm vague on finding eigenvectors and eigenvalues. I thought I'd just ask the question because this specific matrix is of interest to me.

It is an infinite square matrix:

The ones continue on as expected:

other wise:

Essentially I'm trying to understand fractional powers of the matrix, and the only method I am aware of is diagonalization. If that's not possible, does anybody know of any alternative methods? I'm fearful of the matrix logarithm and exponential, but if necessary I could try.

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 Post subject: Re: Can the following matrix be diagonalized?Posted: Thu, 19 Apr 2012 01:07:59 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
JmsNxn92 wrote:
Hello, I'm fairly new to matrix algebra, and I'm iffy on how to properly find diagonalizations, and determining whether a matrix can be diagonalized. I'm vague on finding eigenvectors and eigenvalues. I thought I'd just ask the question because this specific matrix is of interest to me.

It is an infinite square matrix:

The ones continue on as expected:

other wise:

Essentially I'm trying to understand fractional powers of the matrix, and the only method I am aware of is diagonalization. If that's not possible, does anybody know of any alternative methods? I'm fearful of the matrix logarithm and exponential, but if necessary I could try.

Your matrix cannot be diagonalized. As a bit of an important technicality, it should be mentioned that what you have is not literally a matrix at all.

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 Post subject: Re: Can the following matrix be diagonalized?Posted: Thu, 19 Apr 2012 07:13:34 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
JmsNxn92 wrote:
Hello, I'm fairly new to matrix algebra, and I'm iffy on how to properly find diagonalizations, and determining whether a matrix can be diagonalized. I'm vague on finding eigenvectors and eigenvalues. I thought I'd just ask the question because this specific matrix is of interest to me.

It is an infinite square matrix:

The ones continue on as expected:

other wise:

Essentially I'm trying to understand fractional powers of the matrix, and the only method I am aware of is diagonalization. If that's not possible, does anybody know of any alternative methods? I'm fearful of the matrix logarithm and exponential, but if necessary I could try.

Your matrix cannot be diagonalized. As a bit of an important technicality, it should be mentioned that what you have is not literally a matrix at all.

... and, interpreting as , there doesn't exist such that .

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 Post subject: Re: Can the following matrix be diagonalized?Posted: Thu, 19 Apr 2012 18:32:12 UTC
 Member

Joined: Wed, 28 Sep 2011 23:37:54 UTC
Posts: 23
Alright, doesn't surprise me. Thanks for the help.

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 Post subject: Re: Can the following matrix be diagonalized?Posted: Fri, 20 Apr 2012 06:53:38 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
JmsNxn92 wrote:
Alright, doesn't surprise me. Thanks for the help.

Indeed it shouldn't surprise you. There are no square-root of the matrix , for example. The theory doesn't want non-invertibles for a reason.

On the other hand, diagonalisability is something we can get around, at least in finite dimensions, by using the Jordan normal form (infinite-dimensional Jordan normal form is highly sensitive to small perturbations, plus other technical problems like the presence of pseudospectrum, which makes them rather useless in general). For example, has a square-root, namely , which you can obtain from the series , N nilpotent.

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