ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Finding the Eigenvectors of a Matrix:

Updated on April 4, 2013

God's "unconditional" love is discussed at paragraphs 7, 8, 9, 10 and 11 at the end of this hub.

PREPARATION FOR THE FS MATRIX:

1. The arithmetic to find the eigenvectors for the Fibonacci sequence( FS) matrix will be tedious. It is neither hard nor tedious for the 2 X 2 matrix in this hub, but because I want to maintain the exact values for the eigenvector components--i.e. no decimals will be used--of the FS matrix, the algebra will be laborious, even through the FS matrix is also only a 2 X 2 matrix. This hub is an introduction to finding the eigenvectors of the FS matrix. I chose a simple matrix on page 271 from one of my textbooks( Elementary Linear Algebra, 3rd ed. by Paul C Shields). This textbook is a thorough, well written, and understandable introduction to Linear Algebra.

2. If you read the hub on finding the eigenvalues of the FS matrix, you may remember we found the eigenvalues by solving L1, and arriving at the quadratic equation at L2, and getting the 2 eigenvalues given at L2. The eigenvalues for matrix A at L3 are 1 and 7. Matrix X at L3 is an eigenvector for matrix A. If matrix A is multiplied by matrix X, then the product matrix is [7 14]t at L4. Notice an eigenvalue was factored out of the product matrix, which left the eigenvector as a factor. This is one way of knowing that X is an eigenvector because when it is multiplied with its eigenvalue it gives the product matrix at L4. The eigenvectors of a matrix are found by solving the matrix equation at L5.

3. At L6 we plug in the values of A, the project matrix; λ, the eigenvalue; I, the identity matrix, and X, the eigenvector matrix. Then the work is done at L7, L8, L9, and when the two matrices are multiplied at L9 we get the two simultaneous equations at L10. The hub, Solving 4 equations, describes how to use matrices to solve simultaneous equations, and t.elia does a superb job in explaining how to multiply matrices. Before we solve L10, let's see how its solutions makes L5 a true statement.

4. We already know that X is an eigenvector of Matrix A, so let's plug it into L5 to see if it makes that matrix equation a true statement. Matrices can be distributed as at L11, but they are not generally commutative; therefore, you must keep matrix X to the right of each factor since it is being distributed from the right. We will do it both ways: multiply matrix X with the left term of L11, and then distribute it through as on the right side of L11. L8 and L9 shows how we got the 1st matrix at L12, and this is multiplied by X, and it does make L5 a true statement. Next we distribute it through at L13, and do the multiplication at L14 and L15, and once again it proves L5 to be a true statement. We know, then, that X is an eigenvector of matrix A.

5. OK, let's solve the simultaneous equations at L10 by using matrices beginning at L16. These row manipulations are explained at Solving 4 equations( linked at paragraph 3 above). We get the echelon form at L16, which represents the equations at L17. These equations are two vectors on the same line as shown at L18. We can plug any value in for Xj in the 2nd equation of L17 because zero times any number is zero; therefore, the 2nd equation remains true regardless of what number we choose. We will choose 1, and then plug it into the 1st equation of L17, and solve for the 1st variable, Xi. This is done at L19 and L20.

6. We arrive at a value of 1/2 for the 1st variable( vector component), Xi. We can get rid of the fraction by multiplying the entire matrix by 2 at L21, and we get the eigenvector that corresponds to the eigenvalue of 7. Multiplying a vector by a real number does not change its direction, just its length; therefore, we have the same vector( as far as its direction is concerned) whether it is [1/2 1]t or [1, 2]t. L22 and L23 quickly solves for the 2nd eigenvector for matrix A. As I said, this hub is an introduction to solving the eigenvectors of the Fibonacci sequence matrix, and when we get to that we will be more meticulous in doing the math.

IS GOD'S LOVE UNCONDITIONAL?

7. I would like to write a hub on this subject, but for now I'll just give you a few things to ponder.

8. What we want God to be, and what He is, are two different things. What leads many people into eternal hell is assumptions concerning God. We base our lives on these assumptions; thereby thinking our salvation( going to heaven) is secure, when in fact after we die Christ says--to our everlasting horror--" I never knew you: Depart from me . . . "( Matthew 7:23). The reason God gave us His Word, the Bible, is so we would not base our lives on assumptions. We must believe the Bible in its entirety, not just the verses that fit our desires and "needs". For example, Christians find comfort in the fact that God said, " I Am the Lord that healeth thee."( Exodus 15:26), but this verse also has the condition of diligent obedience. Psalm 103:3 also says( as does scores of other verses) that God heals, but additional conditions for this healing is at Isaiah 58:6--14; in fact there are over 15 conditions in these 9 verses.

GOING TO HELL:

9. Do you think it is unconditional love that most( Matthew 20:16; 22:14) people will experience eternal misery and torment in hell? Scripture makes it clear that hell exists( Matthew 8:12; 22:13; 24:51; 25:30; Revelation 14:10, etc), and the context of those verses, and those associated with them is people will spend eternity in hell. The conditions to stay out of hell, and go to heaven is to obey God( Deuteronomy 7:9,10; Micah 6:8; 1 Corinthians 2:9; 6:9,10; Leviticus 18:3,4,and 5; Galatians 6:7,8 and 9). Saving faith( Ephesians 2:8) is having enough faith to do what God said( James 2:20,26). In fact James points out that the devils believe and tremble( 2:19); they believe in God, but not enough to obey Him.

OUR OPINIONS ARE USELESS AND DESTRUCTIVE:

10. Our opinions of who God is, and what He does has no relevance to the Truth, unless they can be backed up with Scripture. God is Love, and He is merciful, but He demands repentance and obedience. To think one can live a lukewarm lifestyle, and still make it to heaven is self deception.

11. This subject of God's "unconditional love" is continued at paragraphs 12, 13, 14, 15, and 16 of hub#12.9( Finding the Eigenvectors of the FS matrix)

Comments

    0 of 8192 characters used
    Post Comment

    • Caleb DRC profile imageAUTHOR

      Caleb DRC 

      6 years ago

      Hi North Wind,

      I appreciate what you said about the math hubs. I always have a sense of excitement when doing the math concerning God's omnipotence and omniscience when He applied, and applies, these qualities of His existence to His creative power. The power of quasars, black holes, angular kinetic energies, and vacuum space is awesome and demonstrates His omnipotence when the numbers are found for them. His omniscience is displayed when the numbers are developed for the complexity of DNA, proteins, and other organic systems. I can't count the times I completed the arithmetic and sat back and said, WOW!!!

      Your last paragraph I agree with 100%. Since I'm still on the "I can't count the times" subject, I can't count the times I figured that out the hard way. If I just would have had faith in God's Word, I could have saved myself a whole lot of troubles, but--oh no!--my opinion is omniscient.

    • North Wind profile image

      North Wind 

      6 years ago from The World (for now)

      I clicked all the buttons except for funny, Caleb DRC. I don't know if I have ever told you but I thoroughly enjoy and admire your Mathematical hubs. I think it is wonderful that you bring it all back to the glory of God as it all should be.

      Our opinions are useless and destructive. Are they not simply vanity? Who are we to think we know more than God and why do we think we can change who He is ? His word is clear and cannot be dismissed. I praise Him for that and I thank you for this.

    working

    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://hubpages.com/privacy-policy#gdpr

    Show Details
    Necessary
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Features
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Marketing
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Statistics
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)