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Finding the Eigenvectors of the Fibonacci Sequence Matrix:

Updated on April 4, 2013


1. The subject of God's "unconditional love" is addressed at paragraphs 12, 13, 14, 15, and 16.


2. At paragraph 2 of hub#12.6( Finding the Eigenvalues of . . . ) it was said that the eigenvalues represent the essence of a matrix. But the essence of something has to be structured within it; otherwise, it would not be the essence of that something. So, in my opinion, the eigenvectors represent the structure upon which the eigenvalues can do their remarkable and outstanding job. The eigenvalues determine the eigenvectors themselves, and the eigenvectors can take their matrix into The Twilight Zone. Hyperspace( multdimensional space), extremely high powers, complex roots and complex powers are all familiar waters for eigenvectors, and the matrices they carry along for the ride. When matrices were first developed they were considered pure mathematics, but eventually they found practical applications in our technologies( big in electronics, for example), and in describing God's creation as in quantum mechanics. Mathematics is God's language, and I believe all pure mathematics will ultimately have some type of applicability to something God has created. Eigenvectors can peer deep into God's creation.


3. As mentioned at paragraph 1 of hub#12.10( Finding the Eigenvectors . . . ) the exact values of each element of the matrices in this hub#12.9 will be maintained in the calculations. Doing this will enable anyone who wants to find the nth term of the FS( Fibonacci Sequence) to any accuracy his/her calculator or computer will allow will be possible. This will make the work more laborious; however, it will also demonstrate how valuable one's basic math skills are when doing something like this. Things like finding a common denominator, rationalizing the denominator, and working with complex fractions( fractions in numerator or denominator or both) become progressively more important as one continues to learn more advanced math.


4. The eigenvalues of λ1 and λ2 were found at hub#12,6( linked at paragraph 2 above). The eigenvectors are found by solving the matrix equation at L1 for λ1, and solving the matrix equation at L2 for λ2. L3 converts L1 into its matrix equivalent. λ1 is multiplied by each element of the identity matrix at L3, and then the product of those multiplications are subtracted from each element of matrix A, and all of this was done at L4, L5 and L6. X1 at L5 is the eigenvector matrix for λ1, and it is represented in matrix form as the right factor in L6. The one on top is Xi( X eye), the one on the bottom is Xj( X jay). Xi is the real component of the vector( the component describe on the x-axis), and Xj is the imaginary component of the vector( the component described on the y-axis). These components are what we are determined to solve, and once we find the values of Xi and Xj for both eigenvectors then we will be done. We could multiply these matrices at L6, and then solve the resultant simultaneous equations( four useful links explaining how to do this are at paragraph 5 of hub#12.1( How to solve 4 . . . ), or we could solve these simultaneous equations by converting the left matrix to echelon form as explained at hub#12.1. The later will be done beginning at L7.


5. Row notation and manipulations were explained beginning at paragraph 8 of hub#12.1; therefore, at L7 the left factor of R1 is multiplied by row 1 of the matrix at L6, and then that product is added to row 2, and then that sum is placed on row 2. Those three dots at the end of L7 mean, "therefore." L8, L9, L10, L11, and L12 all give the details of what L7 instructed us to do. L8 calculates the value of element a1,1 when the multiplication is done. When this value of --1 is added to a2,1 of the left matrix of L6 then we get its new value of zero at L9. L10 begins working on the second column of the left matrix of L6. A common denominator is needed to do the addition at L10; therefore, we rationalize the denominator at L10 and L11. At L12 a negative 2 is factored our of the rationalized denominator of negative 4, which gives us the common denominator; therefore, we can now do the addition, which gives zero as the answer. This is the new value for a2,2. L13 just summarizes what was done from L7 to L13, and it gives the echelon( explained at paragraphs 7, 8, and 11 of hub#12.1) form of the left matrix of L6. At L14 these two matrices are multiplied giving the two simultaneous equations at the right of L14.


6. Any number times zero is zero; therefore, the last equation( lower right) at L14 can have a 1 plugged in for Xj and the equation will remain true. Then we substitute this value of 1 into the Xj of the first equation( upper right) of L14, and that work is done at L15. At L16 we isolate Xi to determine its value. We rationalize the denominator of this value of Xi at L17 and L18. These values of Xi and Xj are the real and imaginary components respectively for the eigenvector, V1, of the eigenvalue of λ1, and this vector is given its matrix from at the end of L18


7. The work from L19 through L32 to find the value of V2, the eigenvector for λ2, is nearly identical to that done for V1, and it would be redundant to go through explaining it again. The matrix P at L33 represents both eigenvectors for the Fibonacci Sequence matrix. We will be using this matrix, P, to determine the diagonal matrix form for matrix A, the Fibonacci Sequence matrix. This diagonal matrix will be the powerful engine we will use to raise the Fibonacci sequence matrix to the trillionth power. L34 and L35 rationalizes the denominator of a1,1 of V2 at L32.


8. The text and I do not have the same eigenvectors; however, they( my vectors and theirs) both work in giving correct answers. In case you do these calculations independent of mine, and you come up with different answers, then do not waste a bunch of time trying to figure out what you did wrongly as I did. Check your answer to see if it predicts accurate values. The main thing is our angles are equal; the vector's magnitude can be different. For example, if we assumed a value of 2, or 7, or 12, or whatever, for Xj at L31 or L15 above then we would have determined vectors of different magnitudes, but their angles would all be respectively equal. You would think I would have thought of that before wasting so much time trying to figure out what I did wrongly, but in my defense I never said I was the sharpest tool in the shed.


9. Another thing of which you need to be aware is the polar form of a vector I used at L39 through L54. The vectors I listed at L51 through L54 are in their polar form, but there is a discrepancy here also. I studied electronics very many years ago, and the form I used at L51 through L54 for those vectors was called the "polar form" of the vector; however, I have studied other textbooks that give polar forms that do not look anything like what I used. So as you progress in your studies do not be surprised if you run into this discrepancy.

10. This hub is not about polar equations; or polar form of functions and vectors; therefore, I have not explained L39 through L54. I just included it so anyone who is interested can have an easy way to compare the vectors. Someday--haven't a clue when--I will do a hub on polar equations. A major advantage of polar equations is in integration. Some equations that are nearly impossible to integrate can be, in some cases, very easy to integrate if the equation is converted to its polar coordinates.


11. L51 through L54 compares our vectors. Notice we arrived at the same angles but we have different lengths or magnitudes. Actually, even their lengths are equal but they are switched.

11.1 So, WHAT IS LEFT? We have to find the inverse of matrix P at L33. Then we have to determine the diagonal matrix that represents the Fibonacci sequence matrix( left matrix at L3), and then that is about it. We can then find the trillionth term of the Fibonacci Sequence.


12. This is a continuation of paragraphs 7, 8, 9, 10 and 11 of hub#12.10 concerning God's "unconditional" love. A sin that would rank toward the top of the list is the sin of misrepresenting God. I'm well aware of this fact as I write these paragraphs concerning God's character. Keep in your mind that I'm not giving you my opinion. I have listed 35 verses of Scripture in the 5 paragraphs of hub#12.10, and I'll be listing additional verses in this hub. If you disagree with the argumentation presented then you must have the Scripture to back up your disagreement, and you must be able to explain away the Scripture I'm using to make my case.


13. The condition to not be blotted out of God's book of life is to not sin, and the condition to be blotted out of God's book of life is to sin. God made this perfectly clear to Moses at Exodus 32: 33. This is also mentioned at Deuteronomy 9:14; 29:20; Psalm 69:28; and Revelation 3:5( notice the condition, " He that overcometh"). Heaven, or hell--there are no other options. We all go to one or the other. Those in hell will never get out; it is a permanent existence of suffering, and no rest. God has given the condition on how to get to heaven: Keep His commandments( 1 John 2: 3,4, 22--25; 3:3--10; 5:18), and Jude makes this unambiguously clear at 1:5, 6, 7!!!, as does Peter at 2 Peter 2:3--6. The remarkable aspect of being blotted out of the book of life is you have to be in it to be blotted out. What possible reasoning can there be to argue that God's love is unconditional when Scripture makes it clear that being blotted out can and does occur?


14. Other assumptions concerning God may be that He does not get angry or He is incapable of Wrath. North Wind wrote an informative hub titled, The Character of God. I suggest you read the whole hub, but the part that is pertinent to this hub is "God and Wrath." She proves with Scripture that God is capable of both anger and wrath( anger bumped up a few notches). After you read that, go to Voice CIW's hub, The Great Day of God's Wrath is Near ( this hub caused me to be one of his followers). I have linked you to these hubs before, but they are both very pertinent to this subject in these paragraphs.


15. God's love is not unconditional, and the subject of God's wrath and hell give more evidence of this: John 3:36( "wrath of God abides on him" if the condition of V36 is not met); Matthew 3:7( "wrath to come") if the condition of V8 is not met; and V10 states the result of God's wrath--hell. Luke says the same thing: 3:7( wrath), V8( condition), V9( result--"cast into the fire"), V10( "what shall we do then", which implies ACTION; i.e. obedience). Romans 1:18( God's wrath and the conditions( sins) that bring it), as homosexual sins( V24, 26, 27), and other sins( V29 to 31) causing God's judgment( V32). Hebrews 3:11 says, "shall not enter into my rest", which is another way of saying they are going to hell. To avoid this the conditions are: hear( V7), harden not your hearts( V8), don't tempt or prove God( V9), don't have an evil heart of disbelief( V12), don't err in heart, or have willful ignorance of God( V10). Revelation 14:10 hits particularly hard, but one can avoid this eternal torment if the condition of V9 is kept( don't take the mark of the beast, and don't worship the beast, or his image).


16. In Rev Earl Jackson's hub, The Unpopular Promises of God, he said the Bible is not a menu. We can not pick and choose what we want to believe. We most definitely want to believe the good promises of the Bible, but we most definitely want to believe the "unpopular" promises of the Bible. In fact we most definitely want to believe the Bible in its entirety. Heaven and Hell are real places, and we will all be in one or the other forever. Now is our probationary time. After we die there are no more chances. It is over! Assumptions concerning God can--and often do--take us straight to hell.

This subject of God's "unconditional love" is continued at paragraph 12 of hub#12.7


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    • North Wind profile image

      North Wind 5 years ago from The World (for now)

      Thanks for the explanation, I understand now. I suppose the mathematics that is considered to be pure are those the mathematicians simply have not figured out how to use as yet. Perhaps because the scientists have not discovered the thing it could be used for? I just find that the two subjects go hand in hand really. God has to be the great Mathematician as well as the great Scientist. :)

    • Caleb DRC profile image

      Caleb DRC 5 years ago

      Rev Earl,

      It will not take much arm twisting to get me to collaborate. I would love to work with you to demonstrate God's omniscience and omnipotence, and to lead others to a saving faith( i.e. obedience) of Jesus Christ. What a great Way to spend our time: leading others to eternal life and away from eternal hell!

      I have release my copyrights in Read Me First so you can do just what you suggested in your comment above. Hub Pages does not want promotion of our hubs in the comments section so I will email you with much more detail. I also have a way you can generate income for your ministry, but since that also will involve promoting my hubs I will include this in your email.

    • Caleb DRC profile image

      Caleb DRC 5 years ago

      Oh, I knew something was up, North Wind, because I have learned that you are as dependable as the sun rising in the eastern sky. I'm just glad that it was your computer, and not something more serious like those pesky aliens coming back to steal some more of your fancy socks.

      I think the general consensus would back up that persons opinion that mathematics is not a science; however, I do not think science would have progressed very far without math, especially some branches as physics. Science is what we use to understand God's creation, but math is the tool which we use to dig deeper and deeper into God's unfathomable works. The more we learn of the complexity and power inherent in God's creation the more math we need to understand it all. As we scrutinize and peer deeper into what God has wrought the equations and mathematical techniques become increasing more complex and abstract. This should not surprise us; well, it doesn't surprise me because I believe God built His universe, through Jesus Christ, upon His mathematics. The air we breath is not considered part of our bodies, and yet we have learned that it is indeed a part of every cell in our bodies, and enables our bodied to build internal structures and to survive. I see math as having an analogous relationship with science.

      Pure math, as I understand it, has no practical application to our lives. It is good for nothing except its inherent beauty. Pure math is no longer considered pure when it finds some sort of practicality to our lives and investigations. The problem with this definition is it does not stay pure very long. Eventually it can be used to investigate something God has created or be applied to some new technology we are developing. I'm no expert on the history of math, but matrices, I understand, began this way. They were fascinating to mathematicians, but that is about it. Perhaps they had application before quantum mechanics, but I know they are used extensively in that realm now. Quantum mechanics digs about as deep as we can get into God's creation, and sure enough the math applied to this is very complex and intricate. The math used in theoretical physics goes beyond even that of quantum mechanics. Again, much of that math was pure, but not anymore.

    • North Wind profile image

      North Wind 5 years ago from The World (for now)

      Hi Caleb DRC,

      Yet another great hub. I am sorry it took so long for me to get to this but I had some computer problems and could not find my way to a working one for about a month. I like to read your hubs when they are 'hot off the press'.

      In paragraph two, you said that matrices were first thought of as pure mathematics. What are they thought of as now? I had a recent conversation with someone concerning Mathematics and I was told that Math simply facilitates Science but really is not a Science. I just thought that was a silly think to say. Are matrices considered to be part Science? What would you say about the whole Math being a Science thing? I'd love to hear your explanation whether it be yay or nay because I know I will be able to understand better.

      Thanks so much for this great hub!

    • Rev Earl Jackson profile image

      Rev. Earl Jackson 5 years ago from Massachusetts

      I am not a scientist, or a mathematician, but this is one of the most amazing demonstrations of God and the eternal realities, that I have ever seen. I was introduced to the Fibonacci sequence years ago when I began studying lay-out and design in art. I knew back then that everything was mathematical, and that it all showed the glory of God, but your amazing work has confirmed it for me once again. I would love for you to write some articles for my Theological Website, because this stuff is amazing and needs to be given the widest possible exposure. God bless you, and thanks again. Contact me if you would like to collaborate or write some stuff for my science and the Bible page. Earl