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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.
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 × ...
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 ...
The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it. The variant of Gaussian elimination that transforms a matrix to reduced row echelon form is sometimes called Gauss–Jordan elimination. A matrix is in column echelon form if its transpose is in
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.
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Hence the Jordan–Chevalley decomposition can be seen as a generalisation of the Jordan normal form, which is also reflected in several proofs of it. It is closely related to the Wedderburn principal theorem about associative algebras , which also leads to several analogues in Lie algebras .
A matrix normal form or matrix canonical form describes the transformation of a matrix to another with special properties. Pages in category "Matrix normal forms" The following 10 pages are in this category, out of 10 total.