Diagonal Matrix

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A Diagonal Matrix is a matrix in which the entries outside the main diagonal or all zero and in which some entries in the main diagonal are not zero.

See: Matrix Main Diagonal.



References

2011

  • http://en.wikipedia.org/wiki/Diagonal_matrix
    • In linear algebra, a diagonal matrix is a matrix (usually a square matrix) in which the entries outside the main diagonal (↘) are all zero. The diagonal entries themselves may or may not be zero. Thus, the matrix D = (di,j) with n columns and n rows is diagonal if:[math]\displaystyle{ d_{i,j} = 0 \mbox{ if } i \ne j \qquad \forall i,j \in \{1, 2, \ldots, n\}. }[/math]

      For example, the following matrix is diagonal: :[math]\displaystyle{ \begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & -3\end{bmatrix} }[/math]

      The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with only the entries of the form di,i possibly non-zero.

      For example: [math]\displaystyle{ \begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & -3\\ 0 & 0 & 0\\ \end{bmatrix} }[/math] or [math]\displaystyle{ \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 0 & 4 & 0& 0 & 0\\ 0 & 0 & -3& 0 & 0\end{bmatrix} }[/math]

      However, in the remainder of this article we will consider only square matrices. Any square diagonal matrix is also a symmetric matrix. Also, if the entries come from the field R or C, then it is a normal matrix as well. Equivalently, we can define a diagonal matrix as a matrix that is both upper- and lower-triangular. The identity matrix In and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.