Finding the Diagonal Matrix Representation of the Fibonacci Sequence Matrix:
1. Paragraphs 8, 9, 10, and 11 is about pursuing your dreams.
THE LITTLE ENGINE THAT COULD:
2. The diagonal matrix representation of a matrix is "The Little Engine That Could"( see paragraph 7 for the story). Could what? Could do it. It is the engine of a matrix. With it a matrix can be raised to a high power, a rational power, or a complex power. Most textbooks teach with 2 by 2 or 3 by 3 matrices. If there are 4 X 4 or 5 X 5 matrices in the textbook then they will usually have a bunch of zeros so that working with them, as raising to a power, is easier. Before the advent of high powered calculators and computers, if your boss asked you to raise a 7 X 7 matrix with no zeros to the 7th power you would probably be tempted to say, "Do it yourself. I quit." Fortunately with the diagonal representation matrix of that 7 X 7 matrix you can raise it to the 7th power without the help of high tech goodies, and you don't have to quit.
DIAGONAL REPRESENTATION MATRIX( DRM) OF THE FIBONACCI SEQUENCE MATRIX:
3. It is derived from the eigenvalues and eigenvectors of the Fibonacci sequence matrix. Notice the diagonal entries of L1 are the eigenvalues we found at Hub#12.6( linked). The DRM is assigned the variable D at L1. It is found by calculating the matrix equation at L1; i.e. multiplying those three matrices. P-1 is the inverse eigenvector matrix( found at hub#12.22--see links below). "A" is the matrix representation of the FS, and was found at hub#12.5( also below). P is the eigenvector matrix of the FS, and was determined at hub#12.9(linked). As in other hubs I will link you to t.elia's hub on multiplying matrices because she did such an outstanding job showing how it is done.
L1 THROUGH L5 EXPLAINED:
4. So throughout this hub we will follow the instructions in t.elia's hub on multiplying matrices. The matrix equation at L1 is P-1AP=D, and the actual matrices are shown at L2. We begin with the first two matrices at L3. Remember that multiplying matrices is not commutative; therefore, order is very important. The arithmetic is done at L3, and completed at L4. The matrix at L4 is multiplyed by the third matrix, P, at L5. L6 is the entire matrix, D, and if you look closely you will see faint lines separating the four elements. L7 simply completes the arithmetic, and simplifies the fractions. L8 gives us the final answer( the final matrix).
AN ATTITUDE IN DOING MATH:
5. The work in this hub may look imposing but it is not even algebra; it is using the basic laws of arithmetic to combine and simplify fractions. I think this hub is a good example of how basic math can be used to lift one onto higher levels of calculations. The basic laws and theorems of mathematics are indispensable as one applies them at higher realms of calculations. The stronger the foundation, the more weight can be built upon it. So also the stronger your foundation is in math( i.e. basic math), the more you can build upon it.
6. "Mathematics is learned in hindsight." This is a true statement. What you learn this year may not be understood for several years later. You just continue to press forward knowing that understanding will follow along and eventually catch up to you. Doing a lot of problems is the best way to learn math. Many times you learn how to do the math and you always get the right answers, but you have not a clue about why you are doing what you are doing, and then . . . click--the light turns on and you understand the concepts behind the procedures, and techniques.
THE LITTLE ENGINE THAT COULD:
7. The Little Engine That Could got its public start in 1902 in a Swedish Journal. It is a children's story about a very long and heavy train asking huge locomotives to pull him up and over a very steep and high mountain. All the big boys had an excuse of why it could not be done. In desperation the train asks a little switch engine if he could handle the job. "I think I can," said the little engine. He immediately gets in front of the train, hooks up and begins to pull while chanting "I think I can; I think I can; I think I can . . . ." As he goes up the mountain he goes slower and slower, but still slowly repeats, "I--think--I--can; I--think--I--can; I--think--I--can; . . . . Near the top he is barely moving, but in a tired and weak voice he whispers, " I . . . think . . . I . . . can; I . . . think . . . I . . . can; I . . . think . . . I . . . can." His dogged determination pays off as he reaches the top and begins the downward side, and he happily says, "I thought I could; I thought I could; I thought I could."
IT IS BETTER TO BE POSITIVE THAN NEGATIVE:
8. What is gained by being negative? Just do not read that question but pause for a moment, and try to answer it. Has anyone who has achieved great or significant goals been negative concerning the pursuit of those goals? The first step toward any dream is taken with a positive attitude--never a negative one. When obstacles are encountered--and they always are--it is the little engine's can-do attitude that goes over them, around them, under them or just kicks them out of the way. The little engine was a fighter. He may not have known that, but that is what he was. He grit his teeth, and said, I'm going to get this done. No one can win a fight by being afraid, and no dream will be pursued if one is afraid. Do you know what fear does? It paralyzes you. You can't fight. The root of fear is doubt. Fear receives its sustenance and strength from doubt. Faith and doubt can not abide in the same house. The dominate one will push out the other, and action determines which one is dominate. If one acts on his fear; i.e. does nothing, then fear becomes the puppet master. If one acts upon his faith, then faith becomes the driving force of success and accomplishment, but it is action that determines who will be victorious. Faith is positive; doubt is negative. Faith is the conduit that was used to give us our skyscrapers, bridges, computers, Olympic winners, great chefs, successful business, and music that bring tears to one's eyes. What has doubt given us? . . . ah . . . well, . . . ah, well ok it has given us . . . well, let me get back to you on that one.
TEARS, BLINDNESS AND ACHIEVEMENT:
9. My best friend tutored the concepts of statistics to a blind student. He calls me up and asks, "How can I get her to "see" a Venn diagram"? I suggested using clay or silly puddy, and empty tuna fish cans. Stop and ponder the obstacles she had to overcome in order to pursue her dream. How can a blind person learn math? Beats me, but she did it. Do you think she is negative? As John Wayne said in Big Jake, "Not hardly."
10. That same friend had another student who started to cry. She said, "I'm never going to get this." This is understandable; she is the mother of three, works a full time job, and is pursuing her dream. She will finish the course this quarter. Do you think she is negative? Let's ask our good friend, John . . . "Not hardly."
11. Just about everything I wrote in the last 4 paragraphs can be backed up with Scripture, but for now I'll leave with you the words of Solomon, "For as he thinketh in his heart, so is he:"( Proverbs 23:7).
Hub#12.5: Transforming the Fibonacci Sequence into a matrix equation:
- Hub #12.5: Transforming the Fibonacci sequence into a matrix equation:
This is the first step in finding the millionth or trillionth term of the Fibonacci sequence. This hub will take you to any term that your calculator can handle, but the trillionth term will require the eigen-stuff of the matrix, as the eigenvalues.
Hub#12.22: Finding the inverse matrix of the Fibonacci sequence matrix.
- Finding the Inverse Matrix of the Fibonacci Sequence Eigenvector Matrix:
The inverse matrix can be used to solve simultaneous equations. Our use of it will be to find the trillionth term of the Fibonacci sequence when we get to the end of this series of hubs.