Matrix M must be an n×n symmetric positive-definite matrix. Returns an n×n matrix B such that B^T×B=M.
CHOL({4, 12, -16; 12, 37, -43; -16, -43, 98}) ⇒
2 6 -8
0 1 5
0 0 3
Returns a design matrix for M. The design matrix has the same number of rows as M. Each column c in M, from left to right, yields a group of columns in the output. For each unique value v in c, from top to bottom, add a column to the output in which v becomes 1 and other values become 0.
PSPP issues a warning if a column only contains a single unique value.
DESIGN({1; 2; 3}) ⇒ {1, 0, 0; 0, 1, 0; 0, 0, 1}
DESIGN({5; 8; 5}) ⇒ {1, 0; 0, 1; 1, 0}
DESIGN({1, 5; 2, 8; 3, 5})
⇒{1, 0, 0, 1, 0; 0, 1, 0, 0, 1; 0, 0, 1, 1, 0}
DESIGN({5; 5; 5})
⇒ (warning)
Returns the determinant of square matrix M.
DET({3, 7; 1, -4}) ⇒ -19
Returns a column vector containing the eigenvalues of symmetric matrix M, sorted in ascending order.
Use CALL EIGEN
(see CALL EIGEN) to compute eigenvalues and
eigenvectors of a matrix.
EVAL({2, 0, 0; 0, 3, 4; 0, 4, 9}) ⇒ {11; 2; 1}
Returns the k×n matrix A that is the generalized inverse of n×k matrix M, defined such that M×A×M=M and A×M×A=A.
GINV({1, 2}) ⇒ {.2; .4}
(approximately)
{1:9} * GINV(1:9) * {1:9} ⇒ {1:9}
(approximately)
M must be a n×m matrix, m ≥ n, with rank n. Returns an n×n orthonormal basis for M, obtained using the Gram-Schmidt process.
GSCH({3, 2; 1, 2}) * SQRT(10) ⇒ {3, -1; 1, 3}
(approximately)
Returns the n×n matrix A that is the inverse of n×n matrix M, defined such that M×A = A×M = I, where I is the identity matrix. M must not be singular, that is, \det(M) ≠ 0.
INV({4, 7; 2, 6}) ⇒ {.6, -.7; -.2, .4}
(approximately)
Returns the pm×qn matrix P that is the
Kroneker product of m×n matrix
Ma and p×q matrix Mb. One may
view P as the concatenation of multiple
p×q blocks, each of which is the scalar
product of Mb by a different element of Ma. For example,
when A
is a 2×2 matrix, KRONEKER(A, B)
is
equivalent to {A(1,1)*B, A(1,2)*B; A(2,1)*B, A(2,2)*B}
.
KRONEKER({1, 2; 3, 4}, {0, 5; 6, 7}) ⇒
0 5 0 10
6 7 12 14
0 15 0 20
18 21 24 28
Returns the rank of matrix M, a integer scalar whose value is the dimension of the vector space spanned by its columns or, equivalently, by its rows.
RANK({1, 0, 1; -2, -3, 1; 3, 3, 0}) ⇒ 2
RANK({1, 1, 0, 2; -1, -1, 0, -2}) ⇒ 1
RANK({1, -1; 1, -1; 0, 0; 2, -2}) ⇒ 1
RANK({1, 2, 1; -2, -3, 1; 3, 5, 0}) ⇒ 2
RANK({1, 0, 2; 2, 1, 0; 3, 2, 1}) ⇒ 3
Ma must be an n×n matrix, with \det(Ma) ≠ 0, and Mb an n×k matrix. Returns an n×k matrix X such that Ma × X = Mb.
All of the following examples show approximate results:
SOLVE({2, 3; 4, 9}, {6, 2; 15, 5}) ⇒
1.50 .50
1.00 .33
SOLVE({1, 3, -2; 3, 5, 6; 2, 4, 3}, {5; 7; 8}) ⇒
-15.00
8.00
2.00
SOLVE({2, 1, -1; -3, -1, 2; -2, 1, 2}, {8; -11; -3}) ⇒
2.00
3.00
-1.00
Given n×k matrix M, returns a \min(n,k)-element column vector containing the singular values of M in descending order.
Use CALL SVD
(see CALL SVD) to compute the full singular
value decomposition of a matrix.
SVAL({1, 1; 0, 0}) ⇒ {1.41; .00}
SVAL({1, 0, 1; 0, 1, 1; 0, 0, 0}) ⇒ {1.73; 1.00; .00}
SVAL({2, 4; 1, 3; 0, 0; 0, 0}) ⇒ {5.46; .37}
Given r×c matrix M and integer scalar k = nk such that 1 ≤ k ≤ \min(r,c), returns the r×c sweep matrix A.
If M_{kk} ≠ 0, then:
A_{kk} = 1/M_{kk}, A_{ik} = -M_{ik}/M_{kk} for i ≠ k, A_{kj} = M_{kj}/M_{kk} for j ≠ k, and A_{ij} = M_{ij} - M_{ik}M_{kj}/M_{kk} for i ≠ k and j ≠ k.
If M_{kk} = 0, then:
A_{ik} = A_{ki} = 0 and A_{ij} = M_{ij}, for i ≠ k and j ≠ k.
Given M = {0, 1, 2; 3, 4, 5; 6, 7, 8}
, then (approximately):
SWEEP(M, 1) ⇒
.00 .00 .00
.00 4.00 5.00
.00 7.00 8.00
SWEEP(M, 2) ⇒
-.75 -.25 .75
.75 .25 1.25
.75 -1.75 -.75
SWEEP(M, 3) ⇒
-1.50 -.75 -.25
-.75 -.38 -.63
.75 .88 .13