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Matrix - Matrix of Reals

Implements overrides for MatrixReal.pm for WeBWorK In general it is better to use MathObjects Matrices (Value::Matrix) in writing PG problem. The answer checking is much superior with better error messages for syntax errors in student entries. Some of the subroutines in this file are still used behind the scenes by Value::Matrix to perform calculations, such as decompose_LR().

```
Method $matrix->_stringify()
-- overrides MatrixReal1 display mode
```

```
(these are deprecated for direct use. Use the covering Methods
provided by MathObject Matrices instead.)
L($matrix) - return matrix L of the LR decomposition
R($matrix) - return matrix R of the LR decomposition
PL($matrix) - return permutation matrix
PR($matrix) - return permutation matrix
Original matrix is PL * L * R *PR = M
```

Obtain the Left Right matrices of the decomposition and the two pivot permutation matrices the original is M = PL*L*R*PR

` Method $matrix->rh_options`

Meant for internal use when dealing with MatrixReal1

```
Method $matrix->trace
Returns: scalar which is the trace of the matrix.
Used by MathObject Matrices for calculating the trace.
Deprecated for direct use in PG questions.
```

```
Method $new_matrix = $matrix->new_from_array_ref ([[a,b,c],[d,e,f]])
Deprecated in favor of using creation tools for MathObject Matrices
```

` Method $matrix->array_ref`

Converts Matrix from an ARRAY to an ARRAY reference.

` Method $matrix->list`

Converts a Matrix column vector to an ARRAY (list).

```
Method $matrix->new_row_matrix
Deprecated -- there are better tools for MathObject Matrices.
```

Create a row 1 by n matrix from a list. This subroutine appears to be broken

```
Method $matrix->proj
Provides behind the scenes calculations for MathObject Matrix->proj
Deprecated for direct use in favor of methods of MathObject matrix
```

```
Method $matrix->proj_coeff
Provides behind the scenes calculations for MathObject Matrix->proj_coeff
Deprecated for direct use in favor of methods of MathObject matrix
```

```
Method $matrix->new_column_matrix
Create column matrix from an ARRAY reference (list reference)
```

```
This method takes an array of column vectors, or an array of arrays,
and converts them to a matrix where each column is one of the previous
vectors.
Method $matrix->new_from_col_vecs
Deprecated: The tools for creating MathObjects Matrices are simpler
```

```
Overrides of MatrixReal which allow use of complex entries
```

```
Function: cp()
Provides ability to use complex numbers.
```

` Method $matrix->copy`

` Method $matrix->conj`

` Method $matrix->transpose`

```
Method $matrix->decompose_LR
Used by MathObjects Matrix for LR decomposition
Deprecated for direct use in PG problems.
```