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Finding the Characteristic Polynomial of a 4 x 4 Matrix

Updated on June 1, 2013

1. Paragraphs 16, 17, 18 and 19 address the sun's rotational energy, and what it declares about God.


2. Paragraph 3 of hub#12.2( linked) said we will raise our 4 x 4 DNA matrix to the 153rd power, a rational power, and a complex power. The first step to be taken in order to get that job done is to determine the characteristic polynomial of our DNA matrix. Within that function is buried the eigenvalues, and those are dug up by finding the roots of the characteristic polynomial. These are the steps taken to raise a 4 x 4 matrix to a complex power: Matrix(A)--->characteristic polynomial(f) of A--->roots(eigenvalues) of f--->eigenvector matrix( P) of A--->diagonal matrix( D) of A--->put it all together.


3. If anyone has an easier way to do this, feel free to link me to the site explaining it and I will edit this hub with that technique. The only way I know how to find the characteristic polynomial( CP) of a 4 x 4 matrix is with expansion of minors( explained by CG( linked) ). There may be other ways to find the determinant, as the one I used at hub#12.2( see par. 2 above), but I chose expansion by minors to find the CP because I know it works. You do it the same way CG explains( very well), but with some modifications in this hub. CG determines the sign coefficient by alternating the signs, which is fine because it works and it is quick; however, I used the formula method, which is explained at paragraph 7 of hub#12.11( linked). There is a very good reason you want to find the sign coefficients this way : YOU CAN KEEP TRACK OF WHERE YOU ARE AND WHAT YOU ARE DOING! With a calculation like this, you can get lost, but the sign-coefficient-formula will tell you with which element you are working, and exactly where you are( column and row intersection). Another difference between this hub and CG's hub is CG is working with a 3 x 3 matrix, but our matrix is 4 x 4; therefore, we kind of iterate the expansion process. With the first expansion at L5--L8 we are left with three 3 x 3 matrices to expand with their cofactors at L9 and L10.


4. At the end of this article I have grouped some of the other pertinent hubs with this one. They will explain some of the work done in this hub, as do the links I've already supplied. The group associated with finding the nth term of the Fibonacci sequence is also good in supplementing this hub.


5. It is hard to believe that what took 6 steps in hub#12.6( linked) has taken 35 steps in this hub. The first 6 steps in hub#12.6 found the characteristic polynomial( CP) of that 2 x 2 matrix( The Fibonacci sequence matrix), and that is all we are doing in this hub. In this hub the concept is the same, and the technique is relatively the same, but the work is laborious. With lambda being subtracted from the diagonal we directly found the determinant in hub#12.6, but in this hub we use expansion of minors to give four 3 x 3 matrices, and each of those require expansion of minors producing three 2 x 2 matrices. Fortunately there is a zero in column 4 of L1, and when each element of a matrix is multiplied by zero then the product matrix is the zero matrix; therefore, that cuts back on our work, and we are left with three 3 x 3 matrices.


7. The DNA matrix is listed at L1. L3 is symbolically represented at L2. Paragraph 8 explains why the determinate at L2 is set to zero. In other words we are to determine the determinate of the last matrix of L3. L4 is just my symbolic way of saying the determinate of the DNA matrix with the eigenvalues included. L5 is suppose to say, Expand De by C4, but I forgot to put in the y for "by." We expand by column 4 because it has a zero. D1 is multiplied by zero; therefore the entire matrix is zero at L6. This means we only have three 3 x 3 matrices to expand instead of four. We will use L6 to explain the other three matrices at L7 and L8. You cover up row 1 and column 4( explained at links), and their intersection is the cofactor of D1. Notice D1 is the submatrix left over after row 1 and column 4 are covered. Then the sign-coefficient is found, which is a negative 1( -1) because -1 raised to the 5th power is -1. The exact same technique is done for the other three matrices. For example, the next element in C4 is 2; therefore, we cover R2 and C4, and the submatrix left over is D2 at L7. A -1 raised to the 6th power at L7 is a positive 1. L8 has the same argumentation for D4 and that of D3 and D2.


8. At hub#12.6 I did not explain why the determinant is set to zero; therefore, let's do so now. The matrix equation, AX = λX, says matrix A multiplied by eigenvector X equals eigenvalue λ times eigenvector X. The eigenvalues we determine must make this matrix equation true. If we algebraically bring the right side over to the left then we have AX - λX = 0. Since we are working with the DNA matrix let's change A to D, while realizing X is an eigenvector matrix of our DNA matrix; therefore, this change give us DX - λX = 0. Now factor out the eigenvector matrix, X, which gives (D - λ)X = 0. If we multiply X by its identity matrix, Ι, then we do not change its value at all. This was explained at paragraph 6 of hub#12.2( linked). This change gives (D - λ)IX = 0. Distribute the identity matrix giving (DI -λI)X = 0. DI is the same as D, so we can rewrite the previous equation as (D - λI)X = 0. Next we can multiply both sides of this equation by the inverse of X, thereby canceling X on the left and putting 0X-1 on the right; therefore, we now have ( D - λI) = 0X-1 . But 0 times the inverse of X is just 0, thereby giving us the final equation of (D - λI) = 0. Now the 0 on the right is actually the zero matrix; therefore, it is a 4 x 4 matrix with all its elements being zero. The determinant of the zero matrix is zero; therefore, the determinant of (D-λI) must be zero also. Now we go to all the work throughout this hub to find the determinant with λ included through the diagonal and we get the final equation at L34. This equation is the equation of the determinant and the determinant is zero; therefore this equation is set to zero. We then find the zeros( roots) of this equation and we will find all 4 values of the eigenvalues.


9. L9 simplified all the cofactors and rewrote the matrices with the simplified cofactors, 2, -1, and (1 - λ). It is important to remember that all these cofactors are carried throughout the entire process to the end. These will be multiplied with the additional cofactors we will find when the three 3 x 3 matrices are expanded at L11 to the end.


10. It has been my experience that the arithmetic of matrices draws mistakes as a light draws moths. I caught most of them--continuously--before finally uploading the calculations to hubpages, but some mistakes made it through, including misspelling polynomial at the end of this hub. I ones I found I've explained in the text; therefore, if something does not make sense in the pictures then the text may explain it, providing I caught it.


11. D1 is the zero matrix because it was multiplied by zero at L6; therefore, D1 can be ignored, which will save us some work. This is why we want to choose rows or columns with zeros to do our expanding. D2 must be expanded so we follow the instructions of the links given. We begin this at L11 and continue through L18 giving the final answer at L19. I chose column 2 of D2 to do the expanding.


12. The same reasoning and arithmetical procedure goes for L25. We begin to expand D3, located at L10. This expansion begins at L20 with column 3 of D3. The process continues through L24, giving the final answer at L25. It says at L25, "Expanded by C4", but it is supposed to be by C3, not C4. Remember that all the coefficients are carried throughout.


13. The last 3 x 3 matrix to be expanded is D4( located at L10 also). The expansion begins at L26 by column 1 of D4, and the process continues through L31 giving us a final answer at L32. Since D4 has three diagonals with lambda( the unknown eigenvalue), along with a diagonal cofactor with lambda we end up with a 4th degree polynomial with lambda as the variable.


14. To get the final equation at L35 we add up the results of L19, L25 and L32. This will give us the determinant of L2 in the form of the characteristic polynomial( CP). We solve the roots of this equation and we have the eigenvalues of our DNA matrix( located at L1). The constant term of CP, which is a negative 10, or --10, is the determinant of the DNA matrix itself. This was calculated at hub#12.2( see par. 8 above for the link). This equation, CP, was solved at hub#12.26( linked), using Newton's method.


15. There are many qualities to God's character as love, judgment, mercy and wrath, but the two qualities to which His creation will testify is His omniscience( unlimited knowledge and wisdom) and omnipotence( unlimited power and authority). The DNA hubs( see hub#9, and hub#9.1 below) begin to address His omniscience. It requires omniscience to construct a DNA molecule, store information within its geometric structure, have other molecules "read" it, and deliver this information to molecules and cells that will process it, and use it to build protein molecules. This aspect of creation is a drop in the oceans of the knowledge and wisdom required to construct this universe including our inner ears, eyes, brains, spiders and their webs. I think many people forget that the laws of creation, as the law of the conservation of angular momentum, are also a part of the creation of Jesus Christ.

16. What about God's omnipotence? Which part of creation do we choose to testify to that? That is a tough choice. Black holes, quasars, supernovas, stretching of space, and the formation of mountains can begin the list. I chose the sun, but I had to narrow it down because many aspects of the sun glorify God as its luminosity, size, temperature, and nuclear processes. What about its spin? It turns on its axis as the earth does. What did Jesus Christ do to get that ball to spin? Well, I don't know what He did to get it to spin, but I can give a rough estimate of how much power it required by God, through Jesus Christ, to get it to rotate.


17. Isaac Newton described many of the laws that God structured into His creation. Newton's first law of motion states, in part, a stationary object requires a force to get it to move. The more massive is an object, the more force required to get it to move. To get star or planet to rotate is to get it to move on its axis. This is easy to do with a tennis ball; just give it a slight torque and it will spin.


18. Driving non stop at 60 mph it would take about 17 days and 7 hours to drive the same distance as that around the equator of the earth, and yet as big as the earth is, one can fit 1,300,000 earths within the volume of the sun. After careful consideration, and pondering all angles of the situation one can conclude that the sun is bigger than a tennis ball. Having said that, to get the sun to rotate at its current rate would require 1.6 x 1036 joules of energy. Our consumption for energy in the United States of America is greater that our production; however, if we assume they are equal then our productivity from all sources would be 1.06 x 1020 joules. So if we committed all the energy we can produce to getting the sun to spin at its current rate then it would take us 1.51 x 1016 years. Even if we were to commit all the energy production of all the world from all sources, it would still take us 3.18 x 1015 years. The irony of this statement is we can't make any energy of our own. All the sources of energy that we convert to our use are from Jesus Christ. Sunlight, wind, geothermal, nuclear, fossil fuels, etc, are from God. We are simply using what God has given to us.


19. Our sun is just one star. God created about 1.5 x 1023 additional stars( see hub#2 below) in our observable universe, and Jesus Christ got all of them spinning on each of their axes also. Additionally there is orbital kinetic energy. Stars orbit their galactic centers, and planets orbit stars, but let's save this for the end of another math hub. The Bible states that Jesus Christ created everything( Colossians 1:16), and the Bible says God created everything( Genesis 1:1, 16). I assume this is in reference to the Trinity; God, Christ and the Holy Spirit are One, and yet separate. I do not fully understand this, but I believe it. Considering the power Jesus Christ displays in creation, it is remarkable that He suffered the cross, and all that it represents, for anyone who is willing to believe in Him---believe being understood to mean obey. This is another thing I do not fully understand, but I believe it.


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    • Caleb DRC profile image

      Caleb DRC 4 years ago

      My mouser is double clicking and leaving double comments again. I said I was going to fix that, didn't I? Uh . . .

      Ok, fine. I'll use this opportunity to say something really profound. So . . . . . . . . uh.

    • Caleb DRC profile image

      Caleb DRC 4 years ago

      Hi North Wind,

      Our lights, cell phones and cars want nothing from us, and they serve us well. People will use them if all they understand is the on/off switch. God, on the other hand, expects us to comply to His Word, and to be 100% committed to Him. Well, now, all of a sudden many people become Albert Einsteins in intelligence, and so they can come up with fantastic excuses why they need not believe in God, and one of those excuses is insufficient understanding. Either love or fear of God must exist before stubborn knees are bent. God can divide the Red Sea, create galaxies, walk on water, and raise the dead, but with some people the only thing for which their knees will bend is sin.

    • Caleb DRC profile image

      Caleb DRC 4 years ago

      Hi North Wind,

      Our lights, cell phones and cars want nothing from us, and they serve us well. People will use them if all they understand is the on/off switch. God, on the other hand, expects us to comply to His Word, and to be 100% committed to Him. Well, now, all of a sudden many people become Albert Einsteins in intelligence, and so they can come up with fantastic excuses why they need not believe in God, and one of those excuses is insufficient understanding. Either love or fear of God must exist before stubborn knees are bent. God can divide the Red Sea, create galaxies, walk on water, and raise the dead, but with some people the only thing for which their knees will bend is sin.

    • North Wind profile image

      North Wind 4 years ago from The World (for now)

      Hey Caleb,

      Some profound statements made in this hub of yours but I have come to expect that of you.

      I may not fully understand electricity but I still turn on the lights when it gets dark. I may not fully understand cell phones but I still use them to communicate with people who are far away from me. Some people do use the argument that since the Trinity cannot be fully explained that God cannot be the Father, Son and Holy Spirit at the same time. They say it is very hard to put stock in something they do not fully understand and yet they do that with other things every single day.

      That is what I do not understand.