jordan form note.pdf - Jordan Canonical Forms Having...

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Jordan Canonical Forms Having studied the importance of diagonalization and knowing that not every square matrix is diagonalizable we answer the question - what is the next best that we can achieve? There are many ways of looking at diagonalization of a matrix. Here we say that a matrix is diagonalizable if and only if there exists a basis of eigen vectors. This happens if and only if the geometric multiplicity of an eigen value is equal to the algebraic multiplicity of that eigen value. (We also know that 0 < geometric mul- tiplicity algebraic multiplicity). Thus corresponding to every eigen value if we can find as many LI eigen vectors as its multiplicity of root of the characteristic equation (i.e. the algebraic multiplicity) then and then only the matrix is diagonal- izable. This cannot be done always as seen from various examples in the tutorial problems. Whenever there exists a basis of eigen vectors then w.r.t this basis the matrix has a diagonal form. In other words there exists a modal matrix P such that P - 1 AP = diag ( λ 1 , λ 2 , · · · , λ n ). It must be noted that the basis should be an ordered basis i.e. first column of P should be an eigen vector corresponding to the first eigen value λ 1 and so on. What can always be done is that we can always find a basis consisting of gener- alized eigen vectors. And the matrix w.r.t. this basis written in some proper order is precisely the Jordan Canonical form. Thus every matrix has a Jordan Canonical form.
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