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Furthermore, the product of an anti-diagonal matrix with a diagonal matrix is anti-diagonal, as is the product of a diagonal matrix with an anti-diagonal matrix. An anti-diagonal matrix is invertible if and only if the entries on the diagonal from the lower left corner to the upper right corner are nonzero. The inverse of any invertible anti ...
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.
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 A {\displaystyle A} with row index specified by i {\displaystyle i} and column index specified by j {\displaystyle j} , these would be entries A i ...
The matrix of the linear map mapping the vector of the entries of a matrix to the vector of a part of the entries (for example the vector of the entries that are not below the main diagonal) See vectorization: Exchange matrix: The binary matrix with ones on the anti-diagonal, and zeroes everywhere else. a ij = δ n+1−i,j: A permutation matrix.
Closed form solutions can be computed for special cases such as symmetric matrices with all diagonal and off-diagonal elements equal [7] or Toeplitz matrices [8] and for the general case as well. [9] [10] In general, the inverse of a tridiagonal matrix is a semiseparable matrix and vice versa. [11]
In linear algebra, an orthogonal diagonalization of a normal matrix (e.g. a symmetric matrix) is a diagonalization by means of an orthogonal change of coordinates. [1]The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on n by means of an orthogonal change of coordinates X = PY.
The determinant of a diagonal matrix is simply the product of all diagonal entries. Such computations generalize easily to A = P D P − 1 {\displaystyle A=PDP^{-1}} . The geometric transformation represented by a diagonalizable matrix is an inhomogeneous dilation (or anisotropic scaling ).
As the peaks at (x,y) in the 2D synchronous spectrum are a measure for the correlation between the intensity changes at x and y in the original data, these main diagonal peaks are also called autopeaks and the main diagonal signal is referred to as autocorrelation signal. The off-diagonal cross-peaks can be either positive or negative. On the ...