Express the given matrix 
as an outer product expansion 
, where 
and 
are the singular values of 
.
Definitions:
Louisiana
A state in the southern region of the United States, known for its distinct French Creole culture, Mardi Gras, and as the birthplace of jazz.
Black Enlistment
Refers to the historic recruitment and incorporation of African American individuals into military services.
Slave States
States in the U.S. before 1865 where the institution of slavery was legally permitted.
Union Army
The land force that fought for the Northern states during the American Civil War, opposing the Confederacy and fighting to preserve the Union and end slavery.
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