# polar: The left polar decomposition. In tensr: Covariance Inference and Decompositions for Tensor Datasets

## Description

`polar` calculates the left polar decomposition of a matrix.

## Usage

 `1` ```polar(X) ```

## Arguments

 `X` A matrix.

## Details

`polar` Takes a matrix X, of dimensions n by p, and returns two matrices P and Z such that X = PZ. P is a symmetric positive definite matrix of dimension n by n and Z is an n by p matrix with orthonormal rows.

## Value

`P` A n by n symmetric positive definite matrix.

`Z` A n by p matrix with orthonormal rows.

Note that `X == P %*% Z`, up to numerical precision.

David Gerard.

## Examples

 ```1 2 3 4 5 6 7 8``` ```X <- matrix(1:6, nrow = 2) polar_x <- polar(X) P <- polar_x\$P Z <- polar_x\$Z P Z trim(Z %*% t(Z)) trim(X - P %*% Z) ```

### Example output

```         [,1]     [,2]
[1,] 3.974074 4.382549
[2,] 4.382549 6.065745
[,1]      [,2]      [,3]
[1,] -0.5510032 0.1361585 0.8233203
[2,]  0.7278247 0.5610652 0.3943058
[,1] [,2]
[1,]    1    0
[2,]    0    1
[,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
```

tensr documentation built on May 2, 2019, 2:32 p.m.