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For a square matrix, the diagonal (or main diagonal or principal diagonal) is the diagonal line of entries running from the top-left corner to the bottom-right corner. [1] [2] [3] For a matrix with row index specified by and column index specified by , these would be entries with =.
The entries form the main diagonal of a square matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a 11 = 9, a 22 = 11, a 33 = 4, a 44 = 10. In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order .
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero.
A square matrix containing zeros in all entries except for the first row, first column, and main diagonal. Band matrix: A square matrix whose non-zero entries are confined to a diagonal band. Bidiagonal matrix: A matrix with elements only on the main diagonal and either the superdiagonal or subdiagonal. Sometimes defined differently, see article.
A square matrix is a matrix with the same number of rows and columns. [5] An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries a ii form the main diagonal of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom ...
In linear algebra, the trace of a square matrix A, denoted tr(A), [1] is the sum of the elements on its main diagonal, + + +.It is only defined for a square matrix (n × n).The trace of a matrix is the sum of its eigenvalues (counted with multiplicities).
An square matrix with entries in a field is called diagonalizable or nondefective if there exists an invertible matrix (i.e. an element of the general linear group GL n (F)), , such that is a diagonal matrix.
In linear algebra, the identity matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.