#### 16.4.2.7 Matrix Algebra Functions

Matrix Function: CHOL (M)

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```
```
Matrix Function: DESIGN (M)

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)
```
Matrix Function: DET (M)

Returns the determinant of square matrix M.

`DET({3, 7; 1, -4}) ⇒ -19`

Matrix Function: EVAL (M)

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}`

Matrix Function: GINV (M)

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)

Matrix Function: GSCH (M)

M must be a n×m matrix, mn, 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)

Matrix Function: INV (M)

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)

Matrix Function: KRONEKER (Ma, Mb)

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```
```
Matrix Function: RANK (M)

Returns the rank of matrix M, an 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```
```
Matrix Function: SOLVE (Ma, Mb)

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```
```
Matrix Function: SVAL (M)

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}```
```
Matrix Function: SWEEP (M, nk)

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```
```