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cofactors of a matrix are found by computing the determinant of certain submatrices. A principal submatrix is a square submatrix obtained by removing certain...

B} are partitioned into submatrices and if the multiplication A B {\displaystyle AB} is carried out treating the submatrices as if they are scalars, but...

cube mesh, assigning every product of two input submatrices to a single processor. The result submatrices are then generated by performing a reduction over...

In mathematics, a balanced matrix is a 0-1 matrix (a matrix where every entry is either zero or one) that does not contain any square submatrix of odd...

In mathematics, a symmetric matrix M {\displaystyle M} with real entries is positive-definite if the real number x T M x {\displaystyle \mathbf {x} ^{\mathsf...

expresses the determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which allows computing a row echelon form...

doi:10.1109/MCISE.2000.814652. Thompson, R. C. (June 1966). "Principal submatrices of normal and Hermitian matrices". Illinois Journal of Mathematics. 10...

Mieghem 2014. Denton et al. 2022. Thompson, R. C. (June 1966). "Principal submatrices of normal and Hermitian matrices". Illinois Journal of Mathematics. 10...

stable if and only if the sequence of determinants of its principal submatrices are all positive. The two procedures are equivalent, with the Routh test...

(Fiedler & Markham 1986). Partition a matrix and its inverse in four submatrices: [ A B C D ] − 1 = [ E F G H ] . {\displaystyle {\begin{bmatrix}A&B\...

} We recursively iterate this division process until the submatrices degenerate into numbers (elements of the ring R {\displaystyle {\mathcal...

sum of minors, which are the determinants of some (n − 1) × (n − 1)-submatrices of B. Specifically, for every i, the Laplace expansion along the ith...

each matrix into four sub-matrices to be multiplied, multiplying the submatrices in a depth-first fashion.[citation needed] In tuning for a specific machine...

restrictions of the determinants of a certain set of n − m {\displaystyle n-m} submatrices of the bordered Hessian. Intuitively, the m {\displaystyle m} constraints...

is split into a family of submatrices. Large submatrices are stored in factorized representation, while small submatrices are stored in standard representation...

applied to the sub-matrix A22, viewed as an operator on Vλ⊥, and its submatrices. Continue this way until the resulting matrix is upper triangular. Since...

any minor of Ω {\displaystyle \Omega } is non-zero. Equivalently, all submatrices of a DFT matrix of prime length are invertible. In signal processing...

known. The conditions are expressed in terms of the ranks of certain submatrices. The Gaussian elimination algorithm for obtaining LU decomposition has...

equations may be block tridiagonal (see block matrix), with smaller submatrices arranged as the individual elements in the above matrix system (e.g....

is to construct such Householder matrices that act upon the principal submatrices of a given matrix [ a 11 a 12 ⋯ a 1 n 0 a 22 ⋯ a 1 n ⋮ ⋱ ⋮ 0 ⋯ 0 x 1...

partitioned matrix is an interpretation of a matrix as being broken down into submatrices which may be more precisely referred to as a collection of row groups...

251=2^{3}+3^{3}+6^{3}=1^{3}+5^{3}+5^{3}.} Every 5 × 5 matrix has exactly 251 square submatrices. Sloane, N. J. A. (ed.). "Sequence A005384 (Sophie Germain primes p:...

The torus can be unrolled by repeating n−1 rows and columns. All n×n submatrices without wraparound, such as the one shaded yellow, then form the complete...

_{r}(q:q:r,:)\end{bmatrix}}} MATLAB colon notation is used here to indicate submatrices, and Ir is the r × r identity matrix. If A ∈ R m 1 × n 1 {\displaystyle...

Thompson, R.C. (1972). "Principal submatrices IX: Interlacing inequalities for singular values of submatrices" (PDF). Linear Algebra and Its Applications...

for small enough submatrices. The conquer part of the algorithm is the unintuitive part. Given the diagonalizations of the submatrices, calculated above...

J\subseteq [n]} with | I | , | J | ≥ r {\displaystyle |I|,|J|\geq r} . Denote submatrices C = L : , J , {\displaystyle C=L_{:,J},} U = L I , J {\displaystyle U=L_{I...

signature of a quadratic form. Sylvester's identity about determinants of submatrices. Sylvester's criterion, a characterization of positive-definite Hermitian...

processors than the matrix elements. We can replace the matrix elements with submatrices, so that every processor processes more values. The scalar multiplication...

used in ADDs corresponds to recursively partitioning the matrix into submatrices. ADDs were first implemented for sparse matrix multiplication and shortest...

where K and F are known and K, x and F are divided into submatrices as shown above. If F2 contains only zeros, and only x1 is desired, K...

SVD of Y {\textstyle \mathbf {Y} } can be partitioned into submatrices, where some submatrices correspond to the signal subspace and some correspond to...

two Kac–Moody algebras in the right hand side are associated with the submatrices of C corresponding to the index sets I1 and I2. An important subclass...

the i-subresultant polynomial is defined in term of determinants of submatrices of the matrix of φ i . {\displaystyle \varphi _{i}.} Let us describe...

considers all the maximal minors, that is the determinants of the square submatrices that have as many rows as the Macaulay matrix. Macaulay proved that the...

Bergman polynomials are the characteristic polynomials for the principal submatrices of the shift operator. Hankel matrix Meurant, Gérard; Sommariva, Alvise...

The class of copositive matrices can be characterized using principal submatrices. One such characterization is due to Wilfred Kaplan: A real symmetric...

four submatrices of roughly equal size, transposing the two submatrices along the diagonal recursively and transposing and swapping the two submatrices above...

The rank function of a linear matroid is given by the matrix rank of submatrices of this matrix, or equivalently by the dimension of the linear span of...

of this greatest common divisor may be expressed as determinants of submatrices of the Sylvester matrix (see Subresultant). Transfer matrix Bézout matrix...

elsewhere. This proves the asserted complexity for matrices such that all submatrices that have to be inverted are indeed invertible. This complexity is thus...

\operatorname {Fitt} _{i}(M)} of M is generated by the minors (determinants of submatrices) of order n − i {\displaystyle n-i} of the matrix a j k {\displaystyle...

matrix A block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements. Boolean matrix A matrix whose entries are...

outcome at the output of the interferometer is related to the permanent of submatrices of a unitary matrix, a boson sampling machine does not allow its estimation...

matrix if: det ( B ) > 0 {\displaystyle \det(\mathbf {B} )>0} for all submatrices B {\displaystyle \mathbf {B} } that can be formed this way. Topics which...

{o}}}&T_{ro}&T_{\overline {ro}}&T_{{\overline {r}}o}\end{bmatrix}}} , and whose submatrices are T r o ¯ {\displaystyle \,T_{r{\overline {o}}}}  : a matrix whose...

Gutin, G.; Mitra, G.; Zverovitch, A. (2004). "Extracting pure network submatrices in linear programs using signed graphs". Discrete Appl. Math. 137 (3):...

there is no need to re-arrange the matrix. We make a matrix of its 2 × 2 submatrices. [ | − 2 − 1 − 1 − 2 | | − 1 − 1 − 2 − 1 | | − 1 − 4 − 1 − 6 | | − 1...

the rank of this larger matrix is still 2. This implies that all 3×3 submatrices have determinant zero, generating four (4 choose 3) plane equations,...

matrices are built; this is normally conducted via a series of overlapping submatrices of size 6x6. Submatrix matches are then reassembled into a final alignment...

extremal conjecture of Zoltán Füredi and Péter Hajnal concerning forbidden submatrices. This approach foreshadowed the subsequent 2004 proof of the Stanley–Wilf...

"an identical rearrangement of rows and columns leaves a set of square submatrices on the principal diagonal and zeros everywhere else." A nearly completely...

Distance Matrix if and only if for all k × k {\displaystyle k\times k} submatrices S {\displaystyle S} of ⁠ Δ {\displaystyle \Delta } ⁠, where ⁠ k ≤ n {\displaystyle...

partitions of the vertices of the graph, of the maximum rank among certain submatrices of the graph's adjacency matrix determined by the partition. They are...

block diagonal matrix whose blocks are 2 × 2 {\displaystyle 2\times 2} submatrices of the form ( 0 t i − t i 0 ) {\displaystyle {\begin{pmatrix}0&t_{i}...

319–336. doi:10.1080/00207216908900046. —— (1970). "On the nonsingular submatrices of the incidence matrix of a graph over the real field". Journal of the...

be represented by a totally unimodular matrix (a matrix whose square submatrices all have determinants equal to 0, 1, or −1). Tutte (1958) proved that...

0/1-matrix by combinatorial 1-rectangles (i.e., non-contiguous, non-convex submatrices, whose entries are all one (see Kushilevitz and Nisan or Dietzfelbinger...

\mathbf {B} _{12}} and B 14 {\displaystyle \mathbf {B} _{14}} are the submatrices defined by the partitioned matrix B = P p , q ( 2 − k n M n h n ) ) 2...

primary challenges of S-FEM is the difficulty in exact integration of the submatrices representing the relationship between global and local meshes (KGL and...