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This (n 1 + ⋯ + n r) × (n 1 + ⋯ + n r) square matrix, consisting of r diagonal blocks, can be compactly indicated as ,, or (,, …,,), where the i-th Jordan block is J λ i,n i. For example, the matrix = [] is a 10 × 10 Jordan matrix with a 3 × 3 block with eigenvalue 0, two 2 × 2 blocks with eigenvalue the imaginary unit i, and a 3 × ...
Example of a matrix in Jordan normal form. All matrix entries not shown are zero. The outlined squares are known as "Jordan blocks". Each Jordan block contains one number λ i on its main diagonal, and 1s directly above the main diagonal. The λ i s are the eigenvalues of the matrix; they need not be distinct.
The Jordan normal form and the Jordan–Chevalley decomposition. Applicable to: square matrix A; Comment: the Jordan normal form generalizes the eigendecomposition to cases where there are repeated eigenvalues and cannot be diagonalized, the Jordan–Chevalley decomposition does this without choosing a basis.
This is also known as the vertex 1-center problem and can be extended to the vertex k-center problem. Finding the center of a graph is useful in facility location problems where the goal is to minimize the worst-case distance to the facility. For example, placing a hospital at a central point reduces the longest distance the ambulance has to ...
Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination. In this case, the term Gaussian elimination refers to the process until it has reached its upper triangular, or (unreduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes ...
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Jordan-Chevalley decomposition — Let : be any operator on the finite-dimensional vector space over the field . Then admits a Jordan-Chevalley decomposition if and only if the minimal polynomial of is a product