Antidiagonal matrix

In the mathematical subfield of linear algebra, the antidiagonal matrix is a square matrix in which all elements outside the counter-diagonal are zero. So it is of the form

${\displaystyle A={\begin{pmatrix}0&\cdots &0&q_{n}\\\vdots &\scriptstyle \cdot ^{\,\scriptstyle \cdot ^{\,\scriptstyle \cdot }}&q_{n-1}&0\\0&\scriptstyle \cdot ^{\,\scriptstyle \cdot ^{\,\scriptstyle \cdot }}&\scriptstyle \cdot ^{\,\scriptstyle \cdot ^{\,\scriptstyle \cdot }}&\vdots \\q_{1}&0&\cdots &0\end{pmatrix}}}$.

Formal definition

A matrix is called antidiagonal if for all with the entry is zero: ${\displaystyle n\times n}$ ${\displaystyle A=(a_{ij})_{i,j=1,\ldots ,n}}$ ${\displaystyle i,j\in \left\{1,\ldots ,n\right\}}$ ${\displaystyle i+j\not =n+1}$ ${\displaystyle (i,j)}$

${\displaystyle i+j\not =n+1\Rightarrow a_{ij}=0}$.

example

An example of an antidiagonal matrix is

${\displaystyle {\begin{pmatrix}0&0&0&0&1\\0&0&0&2&0\\0&0&5&0&0\\0&7&0&0&0\\-1&0&0&0&0\end{pmatrix}}}$.

properties

The determinant of

${\displaystyle A={\begin{pmatrix}0&\cdots &0&q_{n}\\\vdots &\scriptstyle \cdot ^{\,\scriptstyle \cdot ^{\,\scriptstyle \cdot }}&q_{n-1}&0\\0&\scriptstyle \cdot ^{\,\scriptstyle \cdot ^{\,\scriptstyle \cdot }}&\scriptstyle \cdot ^{\,\scriptstyle \cdot ^{\,\scriptstyle \cdot }}&\vdots \\q_{1}&0&\cdots &0\end{pmatrix}}}$

is

${\displaystyle \det(A)=(-1)^{n \choose 2}q_{1}q_{2}\ldots q_{n}.}$

If all are nonzero, then is invertible and the matrix to be inverse is is ${\displaystyle q_{i}}$ ${\displaystyle A}$ ${\displaystyle A}$

${\displaystyle A^{-1}={\begin{pmatrix}0&\cdots &0&{\frac {1}{q_{_{1}}}}\\\vdots &\scriptstyle \cdot ^{\,\scriptstyle \cdot ^{\,\scriptstyle \cdot }}&{\frac {1}{q_{_{2}}}}&0\\0&\scriptstyle \cdot ^{\,\scriptstyle \cdot ^{\,\scriptstyle \cdot }}&\scriptstyle \cdot ^{\,\scriptstyle \cdot ^{\,\scriptstyle \cdot }}&\vdots \\{\frac {1}{q_{n}}}&0&\cdots &0\end{pmatrix}}}$-

The product of two antidiagonal matrices is a diagonal matrix . The product of an antidiagonal matrix with a diagonal matrix (or vice versa) is an antidiagonal matrix.

Antidiagonal matrices are persymmetric .