Finding the eigenvalues of the Fibonacci sequence matrix:
CHRISTIAN HEADS UP:
1. All my math hubs have something about God in their final paragraph(s). In this hub this is at paragraphs 8--15.
EIGENVALUES--THE ESSENCE OF A MATRIX:
2. I was going to say, "Eigenvalues, the blood of a matrix, or the spirit, or the life, or muscle," but none of those are appropriate analogies because matrices are an extremely powerful mathematical tool without using their eigenvalues at all. Essence is the right word here. Let's say you know someone at work or school. You know and like him, he's a hard worker, but that is all you know. Then one unique day you end up spending some time with him( or her, either way), and you discover he has rock solid character, specific beliefs and principles, remarkable attitude, and a contagious sense of humor. You have discovered his essence. He got things prodigiously done at work but you had no idea he had such a spectacular character. Well, that is what eigenvalues, eigenvectors and eigenfunctions are to a matrix. Matrices get a prodigious amount of work done, but when you meet their eigenstuff( not in the dictionary) you end up being silent, pondering and saying, Wow, that is spectacular. Let's say you have a system of springs, and you write their behavior in the form of a matrix. When you calculate the eigenvalues of that matrix, they will represent, or be associated with, the resonant frequency of that system of springs. The same goes for electronic circuits. For certain circuits the eigenvalues will be associated with their resonant frequency. This is true even on an atomic or molecular level.
WHAT DO I KNOW?
3. Well, it turns out, not much. The more I study math, the more I realize that fact. I cannot tell you the essence of the eigenvalues themselves; why they work the way they do, and why they represent many aspects of God's creation. This is true for so many mathematical concepts. I was watching a lecture from Learners TV, or Academic Earth on topology( I think) and the professor said, "Math is learned in hindsight." What he meant is you are not going to understand everything you are learning as you are learning it. Understanding for many concepts will come later as you stretch yourself forward to learn more. So even if we do not understand the essence of eigenvalues, we can still be impressed with their power. We may not understand all the ins and outs of mechanical engineering, but that does not stop us from enjoying our cars, using our cars, and getting some work out of them.
LET 'S GET STARTED:
4. To reduce redundancy, Finding the Determinant . . . , explains the identity matrix at paragraph 6, and the procedure for finding the determinant of a 2 X 2 matrix is at paragraphs 11--13, and locations LE1 to 10 of that hub( toward the end of it).
5. At L1 is matrix A that represents the Fibonacci sequence, and I is its identity matrix. λ is the Greek letter that normally represents eigenvalues in textbooks. If we multiply a constant by a matrix then each element in the matrix is multiplied by that constant. In the case of L3 we multiplied λ( Lambda) by the identity matrix, I, and L4 executes that operation. L5 begins to find the determinant( see paragraph 4 above).
THE EIGENFUNCTION AND EIGENVALUES:
6. The determinant of matrix A is --1( a negative one); however, after doing L3, L4, and L5 we introduced the variable λ ; therefore, when we calculate the determinant with Lambda( λ) included we arrive at L6, which is the eigenfunction of matrix A. We set the determinant to zero at L6. The eigenfunction in this case is simply a quadratic equation; therefore, we can use the quadratic formula to solve it at L7--L10. To an accuracy of 6 decimal places our eigenvalues are 1.618034 and -0.618034. Notice that the 1.618034 answer is the value of the golden ratio. Ushnav( link to his hub) explains how the golden ratio shows up in our bodies, nature and architecture. Josepher( link to his hub) has . . . well, I forgot. The videos and photos were so amazing I forgot what his hub was about, but I know he mentioned the golden ratio, and he has a fascinating link( "Golden ratio") that goes into many mathematical aspects of it.
WHAT IS LEFT TO DO IN FORTHCOMING HUBS?
7. Next we will find the eigenvectors of matrix A , which will give us the "eigenbasis matrix for A." Then we will find the "diagonal matrix representation" for matrix A. A's diagonal representation is the powerful tool we will be using to raise matrix A to the trillionth power; thereby giving us the trillionth term of the Fibonacci sequence(FS). Eventually we will take the 3rd root and 7th root of the Fibonacci sequence matrix, because as I said at paragraph 18 of hub#12.5(Transforming the FS . . . ) this will be very good preparation for what we intend to do with our 4 X 4 DNA matrix( 4X4 is the coefficient part of it) of hub#12.1 Solving 4 Simultaneous . . . . The eigenfunction of our DNA matrix will be a 4th degree equation, and we will solve for its roots using Newton's method. The reason we are finding the trillionth term of the FS, and taking its 3rd and 7th roots of its matrix is to reveal the power of various mathematical techniques as eigenstuff to find solutions to problems that would be impossible to solve without these techniques. I do not think that engineering and science would have progressed very far if not for the use of eigenstuff.
SCIENCE AND MATH PROVIDE ANCHORED ARGUMENTATION FOR CHRISTIANS:
8. Lately I've been getting involved in the forums as, for example, the "Let's do this logically. A theist against atheist. Discussion Plaza" forum. One of my comments from that forum has been copied below:
9. "Every prediction I made in my first comment has occurred, and yet--as I predicted--not a single answer to V7's questions or mine that has not been easily vanquished. I appreciate you and others who have attempted to give argumentation against God as Creator, but neither science nor mathematics can be accessed to achieve that objective. I have presented both scientific and mathematical argumentation and no one will be able to use the same thing to disprove God as Creator. It would be like using science and math to prove why things fall to the ground, and then try to use science and math to prove that the science and math are wrong. For example, I can use mathematical argumentation to prove that DNA has 1 chance in 10^(50,000,000,000) of forming by random processes. How are you going to use the same mathematics to prove me wrong?"
A HOUSE DIVIDED AGAINST ITSELF CANNOT STAND( MT. 12:25; MK. 3:25):
10. God created, by Jesus Christ, this universe, and He created the mathematics upon which the universe is structured, defined and explained. God is the one who said that a house divided against itself cannot stand; therefore, how would it be possible to use His mathematics against Him? I think it is for this reason that I have never seen any atheist or evolutionist attempt to use mathematics to validate evolutionary theory, or to disprove God as the Creator. Furthermore, I have never seen an atheist or evolutionist address or dispute the mathematics that is used to glorify God as the Creator. If anyone knows of a case in which any of this has been done then please link me to it.
A QUESTION NO ATHEIST CAN ANSWER:
11. In the forum mentioned at paragraph 8, Vector 7 asked the question, "Why are atoms structured?" My response to that question has been copied and pasted below.
12. "Brace yourself, Vector 7, you will not get a single answer to any of your questions. Atheists cannot answer any questions concerning how entropy is violated by random processes.
Let's take your first question: Why are atoms structured? I'll answer that. Because God, through Jesus Christ, created them that way. How do I know that? Their existence violates many of our laws of science, and only God can overpower the very laws of physics that He created. Their existence violates the law of the conservation of energy(mass). The law of the conservation of angular momentum asks the question, "How did the electrons begin orbiting the atomic nucleus, esp. so precisely?" The electrons travel 4.877 million miles per hour in the Bohr orbit( radius), which generates an acceleration of 9 X 10^22 meters per second^2. To put this force in perspective, a mass that weighs one pound on earth would weigh( have a force) of 4.6 X 10^18 tons if it was accelerated to the degree that the electron experiences.
13. Richard Feynman said that if the electrostatic charges were not cancelled between 2 grains of sand, 100 feet apart, then the force between them would be 3 million tons. I said 100 million in another forum, or comment, but I used bigger sand and put them closer together. This reveals the balance God designed into the electrostatic charges of protons and electrons. Incidentally, if someone wants to bring up the quantum mechanical electron . . . that's fine. We will have to bump up the speed to 669.6 million miles per hour in the Bohr orbit, and go into the exquisite design that God put into the wave function.
14. Atheists have to explain God's magnificent creation via random processes fighting entropy. If they can't do that, then atheism is a religion based on conjecture; Christianity is based on evidences. I wrote in another forum, or comment on a hub( can't remember which) that the number of permutations within a single DNA molecule is 10^(50,000,000,000). If we fill our universe with sand there would be only 10^91 grains, and make the universe a billion times bigger and we only have 10^100 grains. So how can random processes fight the odds of 1 in 10^(50,000,000,000)?"
THOUSANDS OF YEARS, AND NOT MUCH HAS CHANGED:
15. In the book of Exodus God preformed several unbelievable miracles to deliver the Israelites out of Egypt. AFTER those miracles the Israelites made a golden calf, and said, "These be thy gods, O Israel, which brought thee up out of the land of Egypt"( EX. 32:4). Now thousands of years later we continuously discover more and more of God's magnificence revealed in His creation, and yet ethereal gods of random processes are constructed from the vacuous minds of unbelievers. Atheists have the same attitude concerning Christians; i.e. we are all air-heads. I think Christians and atheists can agree on one thing---one of us is right, and one of us is wrong. Atheists seem to all believe in the power, veracity and validity of math and science; therefore, why doesn't any atheist--any at all--use these two constructs to debunk creationism?
More by this Author
Assumptions about God at par. 12. It is not possible to take the logarithm of a negative number; however, if you are willing to leave the realm of real numbers and enter the field of complex numbers . . . well then, ...
You can easily solve two simultaneous equations without using matrices. Three is harder but it can be done. But 4 or more? Well, then you want to use matrices. Matrices are very powerful mathematical tools, and when...
Going to heaven? begins at par. 10. DEs finds functions. This hub works backwards. You are given a function, and you find a differential equation associated with it.