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
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{\ 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}}}
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Formal definition
A matrix is called antidiagonal if for all with the entry is zero:
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{\ displaystyle n \ times n}
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{\ displaystyle A = (a_ {ij}) _ {i, j = 1, \ ldots, n}}
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{\ displaystyle i, j \ in \ left \ {1, \ ldots, n \ right \}}
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{\ displaystyle i + j \ not = n + 1}
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{\ displaystyle i + j \ not = n + 1 \ Rightarrow a_ {ij} = 0}
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example
An example of an antidiagonal matrix is
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{\ 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}}}
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properties
The determinant of
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{\ 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}}}
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{\ 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
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{\ displaystyle A}
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{\ 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}}}
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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 .
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