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 April 14th, 2019, 02:11 PM #1 Newbie   Joined: Apr 2019 From: turkey Posts: 3 Thanks: 0 eigenvalues and eigenvectors Hi, I have a problem , Give an example to the T ϵ L(R ^ 4) operator with no real eigenvalues. Please explain April 15th, 2019, 02:00 PM   #2
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 T ϵ L(R ^ 4)
Translate! April 15th, 2019, 02:16 PM   #3
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 Originally Posted by mathman Translate!
$T$ is a linear transformation $\mathbb{R}^4 \to \mathbb{R}^4$ April 15th, 2019, 03:50 PM #4 Senior Member   Joined: Oct 2018 From: USA Posts: 102 Thanks: 77 Math Focus: Algebraic Geometry So, where $\lambda$ is the eigenvalue, and $A$ is the transform matrix: $\displaystyle AX=\lambda IX$ $\displaystyle (A-\lambda I)X = 0$ Since X can only be 0, we know that $(A-\lambda I)$ must be rank 4 and therefore $(A-\lambda I)$ is row equivalent to $I$. Therefore you need $A$ to be some matrix that stays rank 4 no matter what you add/subtract from the diagonals. $A= \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ 4 & 0 & 0 & 0 \end{bmatrix}$ Works since no matter what lambda is the matrix won't have any empty rows or columns. In this case $T = AX$ EDIT: Made a mistake in the original, if there are only $1$'s vectors with all equal elements will be eigenvectors. Apologies Thanks from topsquark and matamat19 Last edited by Greens; April 15th, 2019 at 04:20 PM. Reason: typos. Mistake Fixed April 15th, 2019, 04:00 PM #5 Newbie   Joined: Apr 2019 From: turkey Posts: 3 Thanks: 0 thank you so much April 15th, 2019, 04:51 PM #6 Newbie   Joined: Apr 2019 From: turkey Posts: 3 Thanks: 0 I have one more question... Find the T is a linear transformation C3→C3 operator with eigenvalues 6 and 7 so that no matrix is diagonal Tags eigenvalues, eigenvectors Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post integration Elementary Math 1 March 27th, 2018 05:20 AM calypso Linear Algebra 4 June 21st, 2016 09:46 AM hk4491 Linear Algebra 3 March 4th, 2014 02:03 PM bonildo Linear Algebra 7 June 11th, 2012 05:06 PM bonildo Linear Algebra 4 June 8th, 2012 08:02 PM

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