SQL Server MSYMMEIG Function

Updated 2024-03-06 21:03:09.890000

Description

Use the table-valued function MSYMMEIG to return the D and V matrices representing the eigenvalues and eigenvectors of a real symmetric matrix. The input into the function is a string representation of a real symmetric matrix where the columns are separated by commas and the rows are separated by semicolons. MYSMMEIG returns a single row of two columns containing the string representations of the D and V matrices such that:

\textbf{A}=\textbf{VDV}^\text{T}

Syntax

SELECT * FROM [westclintech].[wct].[MSYMMEIG](
   <@A, nvarchar(max),>)

Arguments

@A

A string representation of the real symmetric matrix.

Return Type

table

Remarks

The string representation of @A must only contain numbers, commas (to separate the columns), and semi-colons to separate the rows.

Consecutive commas will generate an error.

Consecutive semi-colons will generate an error.

Non-numeric data between commas will generate an error.

Non-numeric data between semi-colons will generate an error.

To convert non-normalized data to a string format, use the MATRIX2STRING or the MATRIX2STRING_q function.

To convert normalized data to a string format, us the NMATRIX2STRING or the NMATRIX2STRING_q function.

To convert a string result to a table, us the table-valued function MATRIX.

Examples

Example #1

In this example we supply @A as a real symmetric matrix and return the D and V matrices.

--Create a real, symmetric matrix
DECLARE @A as varchar(max) = '5,15,55,225;15,55,225,979;55,225,979,4425;225,979,4425,20515';
--Create variables to store the eigenvalues and the eignevectors
DECLARE @D as varchar(max), @V as varchar(max);
--Put the eigenvalues into the D matrix; the eigenvectors into the V matrix
SELECT @D = D, @V = V FROM wct.MSYMMEIG(@A);
--Return the eigenvalues
SELECT @D as D;
--Return the eigenvectors
SELECT @V as V;
--Only run this SQL to automatically PIVOT the results into the
--traditional row/column matrix presentation
--DECLARE @M as nvarchar(max) = N'SELECT @cols FROM wct.Matrix(@D) d PIVOT(MAX(ItemValue) FOR ColNum in (@cols))p ORDER BY RowNum';
--DECLARE @cols as nvarchar(max);
 
--SET @cols = (SELECT '[' + cast(ColNum as varchar(max)) + ']' FROM wct.MATRIX(@D) WHERE RowNum = 0 ORDER BY colnum FOR XML PATH(''));
--SET @cols = REPLACE(@cols,'][','],[');
--DECLARE @D_pivot as varchar(max) = REPLACE(REPLACE(@M,'@cols',@cols),'@D','''' + @D + '''');
--EXECUTE(@D_pivot);
--DECLARE @V_pivot as varchar(max) = REPLACE(REPLACE(@M,'@cols',@cols),'@D','''' + @V + '''');
--EXECUTE(@V_pivot);

This produces the following result.

Here are the results formatted as a matrix.

Example #2

The cross product of any real matrix is a real symmetric matrix. In this example we generate an m-by-n random matrix, calculate the cross product and return the eigenvalues and eigenvectors.

--Create variables to store the eigenvalues and the eignevectors
DECLARE @D as varchar(max), @V as varchar(max);
--Put the eigenvalues into the D matrix; the eigenvectors into the V matrix
SELECT @D = D, @V = V FROM wct.MSYMMEIG(wct.CROSSPROD(wct.MRAND(25,10),NULL));
--Return the eigenvalues
SELECT @D as D;
--Return the eigenvectors
SELECT @V as V;
--Only run this SQL to automatically PIVOT the results into the
--traditional row/column matrix presentation
--DECLARE @M as nvarchar(max) = N'SELECT @cols FROM wct.Matrix(@D) d PIVOT(MAX(ItemValue) FOR ColNum in (@cols))p ORDER BY RowNum';
--DECLARE @cols as nvarchar(max);
 
--SET @cols = (SELECT '[' + cast(ColNum as varchar(max)) + ']' FROM wct.MATRIX(@D) WHERE RowNum = 0 ORDER BY colnum FOR XML PATH(''));
--SET @cols = REPLACE(@cols,'][','],[');
--DECLARE @D_pivot as varchar(max) = REPLACE(REPLACE(@M,'@cols',@cols),'@D','''' + @D + '''');
--EXECUTE(@D_pivot);
--DECLARE @V_pivot as varchar(max) = REPLACE(REPLACE(@M,'@cols',@cols),'@D','''' + @V + '''');
--EXECUTE(@V_pivot);

This produces the following result. Your results will be different.

Here are the results formatted as a matrix.

See Also

MUPDATE - perform elementwise operations on a matrix