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Defective matrix
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<h2 id="jordan-block">Jordan block</h2> <p>Any nontrivial <a href="/facts/Jordan_matrix/RSDz7mn5">Jordan block</a> of size 2 × 2 {\displaystyle 2\times 2} or larger (that is, not completely diagonal) is defective. (A diagonal matrix is a special case of the Jordan normal form with all trivial Jordan blocks of size 1 × 1 {\displaystyle 1\times 1} and is not defective.) For example, the n × n {\displaystyle n\times n} Jordan block </p> J = [ λ 1 λ ⋱ ⋱ 1 λ ] , {\displaystyle J={\begin{bmatrix}\lambda &1&\;&\;\\\;&\lambda &\ddots &\;\\\;&\;&\ddots &1\\\;&\;&\;&\lambda \end{bmatrix}},} <p>has an <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a>, λ {\displaystyle \lambda } with algebraic multiplicity n {\displaystyle n} (or greater if there are other Jordan blocks with the same eigenvalue), but only one distinct eigenvector J v 1 = λ v 1 {\displaystyle Jv_{1}=\lambda v_{1}} , where v 1 = [ 1 0 ⋮ 0 ] . {\displaystyle v_{1}={\begin{bmatrix}1\\0\\\vdots \\0\end{bmatrix}}.} The other canonical basis vectors v 2 = [ 0 1 ⋮ 0 ] , … , v n = [ 0 0 ⋮ 1 ] {\displaystyle v_{2}={\begin{bmatrix}0\\1\\\vdots \\0\end{bmatrix}},~\ldots ,~v_{n}={\begin{bmatrix}0\\0\\\vdots \\1\end{bmatrix}}} form a chain of generalized eigenvectors such that J v k = λ v k + v k − 1 {\displaystyle Jv_{k}=\lambda v_{k}+v_{k-1}} for k = 2 , … , n {\displaystyle k=2,\ldots ,n} . </p><p>Any defective matrix has a nontrivial <a href="/facts/Jordan_normal_form/M7ezYbnl">Jordan normal form</a>, which is as close as one can come to <a href="/facts/Matrix_diagonalization/y3MkyTCy">diagonalization</a> of such a matrix. </p> <h2 id="example">Example</h2> <p>A simple example of a defective matrix is </p> [ 3 1 0 3 ] , {\displaystyle {\begin{bmatrix}3&1\\0&3\end{bmatrix}},} <p>which has a double <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> of 3 but only one distinct eigenvector </p> [ 1 0 ] {\displaystyle {\begin{bmatrix}1\\0\end{bmatrix}}} <p>(and constant multiples thereof). </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Jordan_normal_form/M7ezYbnl">Jordan normal form</a> – Form of a matrix indicating its eigenvalues and their algebraic multiplicities</li></ul> <h2 id="notes">Notes</h2> <ul><li>Golub, Gene H.; Van Loan, Charles F. (1996), <i>Matrix Computations</i> (3rd ed.), Baltimore: <a href="/facts/Johns_Hopkins_University_Press/Npd5lJTV">Johns Hopkins University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8018-5414-9</li> <li>Strang, Gilbert (1988). <a href="https://archive.org/details/linearalgebraits00stra"><i>Linear Algebra and Its Applications</i></a> (3rd ed.). San Diego: Harcourt. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-970-686-609-7.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Golub & Van Loan (1996, p. 316) - Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 978-0-8018-5414-9 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Golub & Van Loan (1996, p. 316) - Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 978-0-8018-5414-9 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>