Anti-diagonal matrix

Anti-diagonal matrix

In mathematics, an anti-diagonal matrix is a matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗), known as the anti-diagonal.

More precisely, an "n"-by-"n" matrix "A" is an anti-diagonal matrix if the ("i", "j") element is zero for all "i", "j" ∈ {1, …, "n"} with "i" + "j" ≠ "n" + 1.

An example of an anti-diagonal matrix is: egin{bmatrix} 0 & 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 2 & 0 \ 0 & 0 & 5 & 0 & 0 \ 0 & 7 & 0 & 0 & 0 \ -1 & 0 & 0 & 0 & 0 end{bmatrix}.

All anti-diagonal matrices are also persymmetric.

The product of two anti-diagonal matrices is a diagonal matrix. 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-diagonal matrix is also anti-diagonal, as can be seen from the paragraph above. The determinant of an anti-diagonal matrix has absolute value given by the product of the entries on the diagonal from the lower left corner to the upper right corner. However, the sign of this determinant will vary because the one nonzero signed elementary product from an anti-diagonal matrix will have a different sign depending on whether the permutation related to it is odd or even:

More precisely, the sign of the elementary product needed to calculate the determinant of an anti-diagonal matrix is related to whether the corresponding triangular number is even or odd. This is because the number of inversions in the permutation for the only nonzero signed elementary product of any "n" × "n" anti-diagonal matrix is always equal to the "n"th such number.

External links


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • 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… …   Wikipedia

  • Data Matrix — An example of a Data Matrix code, encoding the text: Wikipedia, the free encyclopedia Reading Data …   Wikipedia

  • List of matrices — This page lists some important classes of matrices used in mathematics, science and engineering: Matrices in mathematics*(0,1) matrix a matrix with all elements either 0 or 1. Also called a binary matrix . *Adjugate matrix * Alternant matrix a… …   Wikipedia

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • Список матриц — Структура матрицы Здесь собраны наиболее важные классы матриц, используемые в математике, науке (в целом) и прикладной науке (в частности). Под матрицей понимается прямоугольный массив чисел …   Википедия

  • Trace (linear algebra) — In linear algebra, the trace of an n by n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii represents the entry on the ith row and ith column …   Wikipedia

  • Quantum logic — In mathematical physics and quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account. This research area and its name originated in the 1936 paper by Garrett… …   Wikipedia

  • Mathematics of radio engineering — A complex valued function. The mathematics of radio engineering is a pleasant and very useful subject. This article is an attempt to provide a reasonably comprehensive summary of this almost limitless topic. While the ideas have historically… …   Wikipedia

  • Quantum entanglement — Quantum mechanics Uncertainty principle …   Wikipedia

  • Exchange interaction — In physics, the exchange interaction is a quantum mechanical effect without classical analog which increases or decreases the expectation value of the energy or distance between two or more identical particles when their wave functions overlap.… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”