History of Determinants of the classical test matrices

 Title: Determinants of the classical test matrices-Definition: For a named classical test matrix $A_n$, this table gives the determinant-  $\det(A_n)$ CITE{WikiDeterminant}, where $n$ is the order of the matrix.+Definition: For a named classical test matrix $A_n$, one of the standard matrices+  used to test numerical linear algebra software CITE{MathWorksGallery}, this table+  gives the determinant $\det(A_n)$ CITE{WikiDeterminant}, where $n$ is the order+  of the matrix. Keywords: - determinant

 - Fiedler matrix - Kac-Murdock-Szego matrix+- Kac-Murdock-Szegő matrix - Minij matrix - Grcar matrix

     title: matrix family     display: $A_n$-    constraints: one of the named matrix families listed in the comments whose determinant-      is stored as a row+    constraints: one of the named test matrix families defined in the comments     values:       hilbert: Hilbert matrix

       cauchy: Cauchy matrix       fiedler: Fiedler matrix-      kms: Kac-Murdock-Szego matrix+      kms: Kac-Murdock-Szegő matrix       grcar: Grcar matrix       wilkinson: Wilkinson matrix

     matrix CITE{WikiLehmer}; $A_{ij}=i^{j-1}$ for the Vandermonde matrix CITE{WikiVandermonde};     and $A_{ij}=1/(i+j)$ for the Cauchy matrix, the scalar default case of `gallery("cauchy",n)`-    CITE{MathWorksGallery}. The Fiedler matrix is $A_{ij}=|i-j|$. The Kac-Murdock-Szego-    matrix is $A_{ij}=2^{-|i-j|}$. The Grcar matrix has $-1$ on the subdiagonal and-    $1$ on the diagonal and on the first three superdiagonals.+    CITE{MathWorksGallery}. The Fiedler matrix is $A_{ij}=|i-j|$. The Kac-Murdock-Szegő+    matrix is $A_{ij}=\rho^{|i-j|}$ with $\rho=1/2$, that is, $A_{ij}=2^{-|i-j|}$.+    The Grcar matrix has $-1$ on the subdiagonal and $1$ on the diagonal and on the+    first three superdiagonals.   comment-toolbox-families: The Wilkinson, Clement, Parter, Ris, Lotkin and Riemann-    matrices use the definitions in MATLAB gallery documentation CITE{MathWorksGallery}-    or the Test Matrix Toolbox source CITE{TMTGallery}. The Wilkinson matrix is symmetric-    tridiagonal with off-diagonal entries $1$ and diagonal entries $|(n-1)/2-(i-1)|$.-    The Clement matrix is the default nonsymmetric tridiagonal matrix, with zero diagonal,-    $A_{i,i-1}=n-i+1$ and $A_{i,i+1}=i$. The Parter matrix is $A_{ij}=1/(i-j+1/2)$.+    matrices use `gallery`'s default parameters and the definitions in MATLAB gallery+    documentation CITE{MathWorksGallery} or the Test Matrix Toolbox source CITE{TMTGallery}.+    The Wilkinson matrix is symmetric tridiagonal with off-diagonal entries $1$ and+    diagonal entries $|(n-1)/2-(i-1)|$. The Clement matrix is the nonsymmetric tridiagonal+    matrix with zero diagonal, $A_{i,i-1}=n-i+1$ and $A_{i,i+1}=i$. The Parter matrix+    is $A_{ij}=1/(i-j+1/2)$.   comment-more-toolbox-families: The Ris matrix, also called the Dingdong matrix,     is $A_{ij}=(1/2)/(n-i-j+3/2)$. The Lotkin matrix is the Hilbert matrix with first     row replaced by ones. The Riemann matrix is $B(2:n+1,2:n+1)$ with $B_{ij}=i-1$     if $i\mid j$ and $-1$ otherwise.-  comment-omitted: The Pascal, Minij, Frank and Moler matrices have determinant $1$-    for every listed order; the default Chow matrices are singular for $n>1$; the-    Redheffer determinants are the small-integer Mertens values; and the second-difference-    determinants are $n+1$. Those determinant facts are recorded in formulas rather-    than as rows, because a table of repeated or small integers would make search-    by number less useful.+  comment-omitted: The Pascal matrix CITE{WikiPascal}, Minij, Frank and Moler matrices+    have determinant $1$ for every order; the Chow matrices with `gallery`'s default+    parameters are singular for $n>1$; the Redheffer determinants are values of the+    Mertens function; and the second-difference determinants are $n+1$. Formulas:   formula-characteristic-polynomial: 'The determinant is the signed constant term

     n}(j-i)$.   formula-cauchy: The Cauchy determinant identity $\det(1/(x_i+y_j))=\dfrac{\prod_{i<j}(x_j-x_i)(y_j-y_i)}-    {\prod_{i,j}(x_i+y_j)}$ applies to the Cauchy and Parter matrices, and to the-    Ris matrix after the common factor $1/2$ is taken from each row.+    {\prod_{i,j}(x_i+y_j)}$ applies to the Cauchy matrix with $x_i=i$ and $y_j=j$,+    to the Parter matrix with $x_i=i$ and $y_j=1/2-j$, and to the Ris matrix with+    $x_i=-i$ and $y_j=n-j+3/2$ after the common factor $1/2$ is taken from each row.   formula-fiedler: For the Fiedler matrix, $\det(A_n)=(-1)^{n-1}2^{n-2}(n-1)$.-  formula-kms: For the Kac-Murdock-Szego matrix with $\rho=1/2$, $\det(A_n)=(1-\rho^2)^{n-1}=(3/4)^{n-1}$.+  formula-kms: For the Kac-Murdock-Szegő matrix with $\rho=1/2$, $\det(A_n)=(1-\rho^2)^{n-1}=(3/4)^{n-1}$.   formula-grcar: For the Grcar matrix with three superdiagonals, the determinants     satisfy $g_1=1$, $g_2=2$, $g_3=4$, $g_4=8$, and $g_n=g_{n-1}+g_{n-2}+g_{n-3}+g_{n-4}$

   formula-wilkinson: The Wilkinson determinants satisfy $d_0=1$, $d_1=(n-1)/2$, and     $d_j=a_jd_{j-1}-d_{j-2}$, where $a_j=|(n-1)/2-(j-1)|$ and $d_n=\det(A_n)$.-  formula-small: The omitted default families have $\det(A_n)=1$ for the Pascal, Minij,-    Frank and Moler matrices, $\det(A_n)=0$ for the Chow matrix when $n>1$, $\det(A_n)=M(n)$-    for the Redheffer matrix, and $\det(A_n)=n+1$ for the second-difference matrix.+  formula-small: The omitted `gallery` families with their default parameters have+    $\det(A_n)=1$ for the Pascal, Minij, Frank and Moler matrices, $\det(A_n)=0$ for+    the Chow matrix when $n>1$, $\det(A_n)=M(n)$ for the Redheffer matrix, where $M$+    is the Mertens function CITE{WikiRedheffer}, and $\det(A_n)=n+1$ for the second-difference+    matrix. Programs:   program-sage:

       from sage.rings.rational_field import QQ ++      # Hilbert matrix, n = 3. The constructor calls the lambda with++      # 0-based indices, so i+j+1 here is the 1-based denominator i+j-1.        A = matrix(QQ, 3, 3, lambda i, j: QQ(1) / QQ(i + j + 1))

   relation: stores the polynomials whose signed constant terms are the determinants     listed here+- table: HREF{T329}[Eigenvalues of the classical symmetric test matrices]+  relation: stores the real eigenvalues whose product is the determinant, for the+    symmetric families listed here - table: HREF{T327}[Condition numbers of the classical test matrices]-  relation: stores finite 2-norm condition numbers for nonsingular members of these-    gallery families+  relation: stores finite 2-norm condition numbers for nonsingular members of the+    test matrix families listed here Links:   WikiDeterminant:

   complete: 'no'   complete-note: it holds the listed nontrivial determinant families for $2\leq n\leq30$,-    with only even orders of the Clement matrix because the odd orders are singular,-    making 363 entries-  rigour details: The generator builds each matrix over $\mathbb{Q}$ and computes-    its determinant exactly by fraction-free Bareiss elimination. Before the entries-    were written, `agents/table-build/dry_run.py` computed all 363 entries, checked-    exactness, and measured the longest written value as 968 characters at the Parter-    matrix with $n=30$, with an entries block of 58.8 KB. The determinants were checked-    against the closed forms and recurrences in the formulas, and every listed family-    with $n\leq6$ was also checked by a direct Leibniz determinant over permutations.-    The omitted small families were checked against CITE{formula-small}. The relation-    to HREF{T330}[the characteristic polynomial table] was checked against the stored-    T330 draft values on 138 overlapping rows.+    with only even orders of the Clement matrix because the odd orders are singular;+    the omitted families have determinants that are identically $1$ or $0$, are Mertens+    values, or are $n+1$, so they are recorded in CITE{formula-small} rather than+    as rows+  rigour details: Each matrix is built over $\mathbb{Q}$ and its determinant is computed+    exactly by fraction-free Bareiss elimination, then checked against the formulas;+    a direct Leibniz determinant checks every listed family with $n\leq6$. CITE{formula-characteristic-polynomial}+    was checked against HREF{T330}[the characteristic polynomial table] on the 138+    rows the two tables share, and the omitted small families were checked against+    CITE{formula-small}. Display properties:   number-header: $\det(A_n)$