# lav_matrix: Utility Functions: Matrices and Vectors In lavaan: Latent Variable Analysis

## Description

Utility functions for Matrix and Vector operations.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47``` ```# matrix to vector lav_matrix_vec(A) lav_matrix_vecr(A) lav_matrix_vech(S, diagonal = TRUE) lav_matrix_vechr(S, diagonal = TRUE) # matrix/vector indices lav_matrix_vech_idx(n = 1L, diagonal = TRUE) lav_matrix_vech_row_idx(n = 1L, diagonal = TRUE) lav_matrix_vech_col_idx(n = 1L, diagonal = TRUE) lav_matrix_vechr_idx(n = 1L, diagonal = TRUE) lav_matrix_vechru_idx(n = 1L, diagonal = TRUE) lav_matrix_diag_idx(n = 1L) lav_matrix_diagh_idx(n = 1L) lav_matrix_antidiag_idx(n = 1L) # vector to matrix lav_matrix_vech_reverse(x, diagonal = TRUE) lav_matrix_vechru_reverse(x, diagonal = TRUE) lav_matrix_upper2full(x, diagonal = TRUE) lav_matrix_vechr_reverse(x, diagonal = TRUE) lav_matrix_vechu_reverse(x, diagonal = TRUE) lav_matrix_lower2full(x, diagonal = TRUE) # the duplication matrix lav_matrix_duplication(n = 1L) lav_matrix_duplication_pre(A = matrix(0,0,0)) lav_matrix_duplication_post(A = matrix(0,0,0)) lav_matrix_duplication_pre_post(A = matrix(0,0,0)) lav_matrix_duplication_ginv(n = 1L) lav_matrix_duplication_ginv_pre(A = matrix(0,0,0)) lav_matrix_duplication_ginv_post(A = matrix(0,0,0)) lav_matrix_duplication_ginv_pre_post(A = matrix(0,0,0)) # the commutation matrix lav_matrix_commutation(m = 1L, n = 1L) lav_matrix_commutation_pre(A = matrix(0,0,0)) lav_matrix_commutation_mn_pre(A, m = 1L, n = 1L) # sample statistics lav_matrix_cov(Y, Mu = NULL) # other matrix operations lav_matrix_symmetric_sqrt(S = matrix(0,0,0)) lav_matrix_orthogonal_complement(A = matrix(0,0,0)) lav_matrix_bdiag(...) lav_matrix_trace(..., check = TRUE) ```

## Arguments

 `A` A general matrix. `S` A symmetric matrix. `Y` A matrix representing a (numeric) dataset. `diagonal` Logical. If TRUE, include the diagonal. `n` Integer. When it is the only argument, the dimension of a square matrix. If m is also provided, the number of column of the matrix. `m` Integer. The number of rows of a matrix. `x` Numeric. A vector. `Mu` Numeric. If given, use Mu (instead of sample mean) to center, before taking the crossproduct. `...` One or more matrices, or a list of matrices. `check` Logical. If `check = TRUE`, we check if the (final) matrix is square.

## Details

These are a collection of lower-level matrix/vector related functions that are used throughout the lavaan code. They are made public per request of package developers. Below is a brief description of what they do:

The `lav_matrix_vec` function implements the vec operator (for 'vectorization') and transforms a matrix into a vector by stacking the columns of the matrix one underneath the other.

The `lav_matrix_vecr` function is similar to the `lav_matrix_vec` function but transforms a matrix into a vector by stacking the rows of the matrix one underneath the other.

The `lav_matrix_vech` function implements the vech operator (for 'half vectorization') and transforms a symmetric matrix into a vector by stacking the columns of the matrix one underneath the other, but eliminating all supradiagonal elements. If diagonal = FALSE, the diagonal elements are also eliminated.

The `lav_matrix_vechr` function is similar to the `lav_matrix_vech` function but transforms a matrix into a vector by stacking the rows of the matrix one underneath the other, eliminating all supradiagonal elements.

The `lav_matrix_vech_idx` function returns the vector indices of the lower triangular elements of a symmetric matrix of size n, column by column.

The `lav_matrix_vech_row_idx` function returns the row indices of the lower triangular elements of a symmetric matrix of size n.

The `lav_matrix_vech_col_idx` function returns the column indices of the lower triangular elements of a symmetric matrix of size n.

The `lav_matrix_vechr_idx` function returns the vector indices of the lower triangular elements of a symmetric matrix of size n, row by row.

The `lav_matrix_vechu_idx` function returns the vector indices of the upper triangular elements of a symmetric matrix of size n, column by column.

The `lav_matrix_vechru_idx` function returns the vector indices of the upper triangular elements of a symmetric matrix of size n, row by row.

The `lav_matrix_diag_idx` function returns the vector indices of the diagonal elements of a symmetric matrix of size n.

The `lav_matrix_diagh_idx` function returns the vector indices of the lower part of a symmetric matrix of size n.

The `lav_matrix_antidiag_idx` function returns the vector indices of the anti diagonal elements a symmetric matrix of size n.

The `lav_matrix_vech_reverse` function (alias: `lav_matrix_vechru_reverse` and `lav_matrix_upper2full`) creates a symmetric matrix, given only upper triangular elements, row by row. If diagonal = FALSE, an diagonal with zero elements is added.

The `lav_matrix_vechr_reverse` (alias: `lav_matrix_vechu_reverse` and `lav_matrix_lower2full`) creates a symmetric matrix, given only the lower triangular elements, row by row. If diagonal = FALSE, an diagonal with zero elements is added.

The `lav_matrix_duplication` function generates the duplication matrix for a symmetric matrix of size n. This matrix duplicates the elements in vech(S) to create vec(S) (where S is symmetric). This matrix is very sparse, and should probably never be explicitly created. Use one of the functions below.

The `lav_matrix_duplication_pre` function computes the product of the transpose of the duplication matrix and a matrix A. The A matrix should have n*n rows, where n is an integer. The duplication matrix is not explicitly created.

The `lav_matrix_duplication_post` function computes the product of a matrix A with the duplication matrix. The A matrix should have n*n columns, where n is an integer. The duplication matrix is not explicitly created.

The `lav_matrix_duplication_pre_post` function first pre-multiplies a matrix A with the transpose of the duplication matrix, and then post multiplies the result again with the duplication matrix. A must be square matrix with n*n rows and columns, where n is an integer. The duplication matrix is not explicitly created. multiplies a matrix A with the

The `lav_matrix_duplication_ginv` function computes the generalized inverse of the duplication matrix. The matrix removes the duplicated elements in vec(S) to create vech(S). This matrix is very sparse, and should probably never be explicitly created. Use one of the functions below.

The `lav_matrix_duplication_ginv_pre` function computes the product of the generalized inverse of the duplication matrix and a matrix A with n*n rows, where n is an integer. The generalized inverse of the duplication matrix is not explicitly created.

The `lav_matrix_duplication_ginv_post` function computes the product of a matrix A (with n*n columns, where n is an integer) and the transpose of the generalized inverse of the duplication matrix. The generalized inverse of the duplication matrix is not explicitly created.

The `lav_matrix_duplication_ginv_pre_post` function first pre-multiplies a matrix A with the transpose of the generalized inverse of the duplication matrix, and the post multiplies the result again with the transpose of the generalized inverse matrix. The matrix A must be square with n*n rows and columns, where n is an integer. The generalized inverse of the duplication matrix is not explicitly created.

The `lav_matrix_commutation` function computes the commutation matrix which is a permutation matrix which transforms vec(A) (with m rows and n columns) into vec(t(A)).

The `lav_matrix_commutation_pre` function computes the product of the commutation matrix with a matrix A, without explicitly creating the commutation matrix. The matrix A must have n*n rows, where n is an integer.

The `lav_matrix_commutation_mn_pre` function computes the product of the commutation matrix with a matrix A, without explicitly creating the commutation matrix. The matrix A must have m*n rows, where m and n are integers.

The `lav_matrix_cov` function computes the sample covariance matrix of its input matrix, where the elements are divided by N (the number of rows).

The `lav_matrix_symmetric_sqrt` function computes the square root of a positive definite symmetric matrix (using an eigen decomposition). If some of the eigenvalues are negative, they are silently fixed to zero.

The `lav_matrix_orthogonal_complement` function computes an orthogonal complement of the matrix A, using a qr decomposition.

The `lav_matrix_bdiag` function constructs a block diagonal matrix from its arguments.

The `lav_matrix_trace` function computes the trace (the sum of the diagonal elements) of a single (square) matrix, or if multiple matrices are provided (either as a list, or as multiple arguments), we first compute their product (which must result in a square matrix), and then we compute the trace; if `check = TRUE`, we check if the (final) matrix is square.

## References

Magnus, J. R. and H. Neudecker (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

## Examples

 ```1 2 3 4 5 6 7 8 9``` ```# upper elements of a 3 by 3 symmetric matrix (row by row) x <- c(30, 16, 5, 10, 3, 1) # construct full symmetric matrix S <- lav_matrix_upper2full(x) # compute the normal theory `Gamma' matrix given a covariance # matrix (S), using the formula: Gamma = 2 * D^{+} (S %x% S) t(D^{+}) Gamma.NT <- 2 * lav_matrix_duplication_ginv_pre_post(S %x% S) Gamma.NT ```

### Example output

```This is lavaan 0.5-23.1097
lavaan is BETA software! Please report any bugs.
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 1800  960  300  512  160   50
[2,]  960  556  170  320   98   30
[3,]  300  170   55   96   31   10
[4,]  512  320   96  200   60   18
[5,]  160   98   31   60   19    6
[6,]   50   30   10   18    6    2
```

lavaan documentation built on March 10, 2021, 5:05 p.m.