DISCOVERING MATHEMATICS WITH MAPLEAn Interactive Exploration for Mathematicians, Engineers, and EconometriciansChapter 6. Vector Spaces and Linear MappingsWorksheet 6b. Eigenvectors and EigenspacesThe topic eigenvalues and eigenspaces is often seen as the culmination of a first introduction to Linear Algebra. Here many important concepts and principles of Linear Algebra meet, and the tools developed up till then are sufficiently powerful to prove theorems on the diagonalization of matrices and of linear mappings. This final worksheet VectW6b.mw is about the way Maple can assist us in finding eigenvalues and eigenspaces of matrices. Naturally, the characteristic polynomial of a matrix will be included as well as the diagonalization process of symmetric matrices. Finally we shall take a look at the computation of the singular value decomposition of a non-diagonalizable matrix.Computing Eigenvalues and EigenvectorsAs usual we begin withLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The following procedure creates a class of special matrices:PkkiZkc2ImYqNiRJImlHRiRJImpHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoOSQiIiI5JUYuISIiRi5GJEYkRiQ=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=For each natural number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , the command Matrix(n,n,f) gives a nice, symmetric matrix.We shall first work with the simplest matrix of this collection.PkkiQUc2Ii1JJ01hdHJpeEdGJDYlIiIjRihJImZHRiQ=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The eigenvalues of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are the solutions to the characteristic equation of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:QyQtSTlDaGFyYWN0ZXJpc3RpY1BvbHlub21pYWxHNiI2JEkiQUdGJUkieEdGJSIiIg==LUkmc29sdmVHNiI2JEkiJUdGJEkieEdGJA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The command CharacteristicPolynomial gives the characteristic polynomial in a variable of our own choice.The eigenvectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= corresponding to the eigenvalues obtained above can be calculated directly using the EigenVectors command:LUktRWlnZW52ZWN0b3JzRzYiNiNJIkFHRiQ=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Maple returns a sequence consisting of a Vector of eigenvalues and a Matrix whose columns are the corresponding eigenvectors in the same order. These results can also be obtained by clicking the right mouse button on the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0Yv (Maple output) and selecting Characteristic Polynomial, Eigenvalues or Eigenvectors from the pop-up menu.The next example, also taken from the collection of matrices defined earlier, is less simple.PkkiQkc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYlIiImRitJImZHRiQ=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The characteristic polynomial of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= --- or, equally, the characteristic polynomial of a linear mapping T: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJWRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlFGMUY+RitGPg== with 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, which is represented by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= relative to some basis of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiNUYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE= --- is usually obtained by evaluating det(x Id - B), where Id is the identity matrix. QyQsJiomSSJ4RzYiIiIiLUkvSWRlbnRpdHlNYXRyaXhHRiY2IyIiJkYnRidJIkJHRiYhIiJGJw==QyQtSTVDaGFyYWN0ZXJpc3RpY01hdHJpeEc2IjYkSSJCR0YlSSJ4R0YlIiIiQyQ+SShjaGFycG9sRzYiLUklc29ydEclKnByb3RlY3RlZEc2Iy1JLERldGVybWluYW50R0YlNiNJIiVHRiUiIiI=LUkmZXZhbGJHJSpwcm90ZWN0ZWRHNiMvSShjaGFycG9sRzYiLUk5Q2hhcmFjdGVyaXN0aWNQb2x5bm9taWFsR0YoNiRJIkJHRihJInhHRig=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Meanwhile, applying sort, we ordered the terms of the characteristic polynomial by decreasing degree. The eigenvalues and corresponding eigenvectors are once more obtained byQyQ+NiRJJ2xhbWJkYUc2IkkjRUVHRiYtSS1FaWdlbnZlY3RvcnNHRiY2I0kiQkdGJiIiIg==QyRJJ2xhbWJkYUc2IiIiIg==QyUtSSZldmFsZkclKnByb3RlY3RlZEc2I0kiJUc2IiIiIi1GJDYjSSNFRUdGKA==QystSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYlL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGLDYlSSJCR0YsSSNFRUdGLEkldHJ1ZUdGJSIiIi1JL0RpYWdvbmFsTWF0cml4R0YsNiNJJ2xhbWJkYUdGLEY0LUYkNiMtRig2JUYySSIlR0YsRjNGNC1GJDYjLCZGJ0Y0LUYoNiVGMkY1RjMhIiJGNC1JJG1hcEdGJTYlSShmbm9ybWFsR0YuRj0iIio=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=From Maple's output we see that the eigenvalue 0 is a multiple zero of the characteristic polynomial of multiplicity 3. We also notice that a basis of eigenvectors corresponding to this eigenvalue comprises three vectors. The eigenspace corresponding to the eigenvalue LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= is by definition equal to the null space of the matrix (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYyId - B). Let us check this now. QyQ+SSdFaWdlblNHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSpOdWxsU3BhY2VHRiU2IywmKiY5JCIiIi1JL0lkZW50aXR5TWF0cml4R0YlNiMiIiZGMkYySSJCR0YlISIiRiVGJUYlRjI=QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJ0VpZ2VuU0c2IjYjLCYjIiNEIiIjIiIiLUklc3FydEc2JEYlSShfc3lzbGliR0YpNiMiI0wjIiImRi5GLw==QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJ0VpZ2VuU0c2IjYjLCYjIiNEIiIjIiIiLUklc3FydEc2JEYlSShfc3lzbGliR0YpNiMiI0wjISImRi5GLw==LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSdFaWdlblNHNiI2IyIiIQ==[Maple Update:
The Nullspace command acting on the negative eigenvalue is not giving the corresponding eigenvector! One sees that the difference of the large numbers in the exact expression below is ending up lost due to the 10 digit default. Increasing the default Digits to 20 fixes this! We use a trick to extract the eigenvector corresponding to the single negative eigenvalue:Qyc/KEkiaUc2IiIiIkYmIiImSSV0cnVlRyUqcHJvdGVjdGVkR0AlMi1JJmV2YWxmR0YpNiMmSSdsYW1iZGFHRiU2I0YkRiZDJD5JLGxhbWJkYW1pbnVzR0YlRi8+SShFRW1pbnVzR0YlLUknQ29sdW1uR0YlNiRJI0VFR0YlRidGJUYmRjYhIiItRi02I0kiJUdGJQ==SSxsYW1iZGFtaW51c0c2Ig==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUShFRW1pbnVzRidGL0YyL0YzUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGZ24vJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRlk2JkZlbkZobkZbby9GXm9RJWF1dG9GJ0YrLUY2NiQtRiM2Ji1GNjYkLUYjNiYtRiw2I1EhRictRiM2KC1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSMyNUYnRj0tRmVwNiRRIjJGJ0Y9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZgcS8lKWJldmVsbGVkR0ZFLUZANi1RKiZ1bWludXMwO0YnRj1GQy9GR0ZFRkhGSkZMRk5GUC9GU1EsMC4yMjIyMjIyZW1GJy9GVkZqcS1GYnA2KC1GZXA2JFEiNUYnRj1GaHBGW3FGXnFGYXFGY3EtRkA2LVExJkludmlzaWJsZVRpbWVzO0YnRj1GQ0ZocUZIRkpGTEZORlBGUi9GVkZULUkmbXNxcnRHRiQ2Iy1GZXA2JFEjMzNGJ0Y9Rj1GXHBGPUY9LUZANi1RIn5GJ0Y9RkNGaHFGSEZKRkxGTkZQRlJGZHJGOkY9Rj1GP0ZYRmBvRistRjY2JC1GIzYqLUYsNiVRIkJGJ0YvRjJGW3NGOi1GQDYtUSgmbWludXM7RidGPUZDRmhxRkhGSkZMRk5GUEZpcUZbckZob0Zbc0Y6Rj1GPUY/LyUrZXhlY3V0YWJsZUdGRUY9QyUtSSpOdWxsU3BhY2VHNiI2IywmKiYsJiMiI0QiIiMiIiItSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMiI0wjISImRixGLS1JL0lkZW50aXR5TWF0cml4R0YlNiMiIiZGLUYtSSJCR0YlISIiRi0tSSZldmFsZkdGMTYjSSIlR0YlNotice that the first three components of the basis vector have large ugly radical differences which evaluate to zero at 10 digits but not at 20!QyctSSZldmFsZkclKnByb3RlY3RlZEc2IywmIS4hKVtxRypmSSIiIi1JJXNxcnRHNiRGJUkoX3N5c2xpYkc2IjYjIiNMIi1QakBsRWBGKS1GJDYjLCYhLVZBV24iPSlGKUYqIi1EMG5DQzlGKS1GJDYjLCYhLFwqZlNWTUYpRioiK1QvPyUqZg==QzA+SSdEaWdpdHNHNiIiIz8hIiItSSZldmFsZkclKnByb3RlY3RlZEc2IywmIS4hKVtxRypmSSIiIi1JJXNxcnRHNiRGKkkoX3N5c2xpYkdGJTYjIiNMIi1QakBsRWBGLi1GKTYjLCYhLVZBV24iPSlGLkYvIi1EMG5DQzlGLi1GKTYjLCYhLFwqZlNWTUYuRi8iK1QvPyUqZkYuLUkqTnVsbFNwYWNlR0YlNiMsJiomLCYjIiNEIiIjRi5GLyMhIiZGSEYuLUkvSWRlbnRpdHlNYXRyaXhHRiU2IyIiJkYuRi5JIkJHRiVGJ0YnLUYpNiNJIiVHRiVGLj5GJCIjNUYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This is an important lesson. Numerical matrix operations can go terribly wrong because of truncation errors and approximate zeros.]DiagonalizationAll this means that for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiNUYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE= a basis exists of eigenvectors of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, because all five eigenvectors that turned up are independent. Hence LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is diagonalizable but we knew that already because LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is symmetric and all symmetric matrices have this property.The matrix EE provides us with sufficient information to verify that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is indeed similar to the diagonal matrix of its eigenvalues.The similarity between B and the diagonal matrix of its eigenvalues is a direct consequence of the result of the following instruction: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSIlRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZAWe used the Maple procedure normal with the option expanded in combination with the map command to simplify the expression.The matrix EE is not orthogonal because LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEjRUVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRi82JVEidEYnRjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGLi9GNlEnbm9ybWFsRic= is not equal to the identity matrix.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEkbWFwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2KS1GLDYlRkFGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4zMzMzMzMzZW1GJy1GIzYnLUYsNiVRNU1hdHJpeE1hdHJpeE11bHRpcGx5RidGNkY5RjwtRlc2JC1GIzYoRistRiM2Jy1GLDYlUSpUcmFuc3Bvc2VGJ0Y2RjlGPC1GVzYkLUYjNiUtRiw2JVEjRUVGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZicEZARmduRl9wRmJwRkBGQEZicEZARmduLUYsNiVRKWV4cGFuZGVkRidGNkY5RmJwRkBGQEZicEZARitGYnBGQEYrRmJwRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Before continuing, take a look at assignments 1 and 2 of worksheet VectA6b.mw. The purpose of assignment 1 is to diagonalize a given matrix, and in assignment 2 an example is given of the use of matrix diagonalization.The Gram-Schmidt Orthogonalization ProcessThe matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is not orthogonal, but we can orthogonalize the eigenspace corresponding to the eigenvalue 0 by means of the Gram-Schmidt orthogonalization procedure. Recall that eigenvectors corresponding to different eigenvalues are automatically orthogonal. However, as we do not know the exact order in which the Maple procedure eigenvects will put the eigenvalues and corresponding eigenvectors, we shall apply the Gram-Schmidt procedure to the union of all three eigenspaces. The commands map and normal are used once more to get simplified and readable expressions for all elements of the list.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=We have not finished yet because the basis of eigenvectors still has to be made orthonormal, that is the length of each vector has to be normalized to 1. The map command enables us to simultaneously divide each individual vector by its own length. Recall that the length of a vector is equal to the value of its Euclidean norm (or 2-norm). So a new list is created of mutually orthogonal eigenvectors of B of length 1.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This looks rather ominous because of the many radicals.On putting the vectors of eigenlist_orthonormal side by side, thus forming the columns of a matrix, we create the orthogonal matrix Q. We first verify Q's orthogonality.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Finally we see that the matrixLambda := map(normal,
MatrixMatrixMultiply(Transpose(Q),MatrixMatrixMultiply(B,Q)),expanded);is diagonal. The next instruction verifies that the diagonal elements of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpJkxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= are the zeros of the characteristic polynomial, and hence the eigenvalues of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The eigenvalues of square, non-symmetric, real matrices are by definition real, in as far as they exist of course. If we consider such a real matrix over the field of complex numbers, then all becomes more transparent. This is so, because the eigenvalues are the zeros of the characteristic polynomial, and the characteristic equation has exactly as many complex roots as its degree, counting multiplicities. The Maple system is perfectly capable of performing computations with complex numbers, and therefore eigenvalues and (complex) eigenvectors can be found for non-symmetric matrices as well.Singular Value Decomposition (SVD)We conclude with an example of a non-diagonalizable matrix.The defining process for this matrix proceeds as it were in reverse order: we start with a given polynomial and then search for a matrix having this polynomial for its characteristic polynomial.The Maple command companion helps us to do this. The result is the so-called 'companionmatrix' of a polynomial with leading coefficient 1. The meaning of the word companion is revealed when we look at the final column of the companion matrix: it is formed by the opposites of the coefficients of the given polynomial, only the leading coefficient is left out.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=In order to test the companion relation between polynomial p and matrix F, we also determine the characteristic polynomial of F in the usual way, that is by means of det(x Id - F).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This seems to be alright. The matrix F is not diagonalizable because both eigenspaces of the multiple eigenvalues 0 and 1 are of dimension 1. Therefore, the number of independent eigenvectors falls short of the number needed to constitute a basis for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiNUYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=. We shall take a closer look at this now.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEtRWlnZW52ZWN0b3JzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJGRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARkBGaG5GQEYrRmhuRkBGK0ZobkZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Just as expected, the output is disappointing. On the other hand, both matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImxGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== := F LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJ0RidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInJGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== := LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJ0RidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= F are symmetric and hence diagonalizable. Let the columns of the matrix U be the orthonormal eigenvectors of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImxGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== and let V be formed likewise by the orthonormal eigenvectors of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInJGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==. Then F can be expressed as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 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 , where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJlNpZ21hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== is the diagonal matrix of singular values. The singular values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are the square roots of the eigenvalues of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImxGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInJGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== (both matrices have the same eigenvalues).The command Singularvalues(F,U,V) can be used to compute the singular values and the matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Round-off errors cause small deviations from the original matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The reason for these errors is that the SingularValues procedure gives only numerical approximations, not exact values.Assignment 3 of worksheet VectA6b.mw is about the SVD of another non-diagonalizable matrix.This ends worksheet VectW6b.mw.