Statement of The Theorem
Suppose M is an m×n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers. Then there exists a factorization of the form
where U is an m×m unitary matrix over K, the matrix Σ is an m×n diagonal matrix with nonnegative real numbers on the diagonal, and the n×n unitary matrix V* denotes the conjugate transpose of the n×n unitary matrix V. Such a factorization is called the singular value decomposition of M.
The diagonal entries of Σ are known as the singular values of M. A common convention is to list the singular values in descending order. In this case, the diagonal matrix Σ is uniquely determined by M (though the matrices U and V are not).
Read more about this topic: Singular Value Decomposition
Famous quotes containing the words statement of the, statement of, statement and/or theorem:
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in one’s theme, it needs no translation into art, the bold statement of it being sufficient.”
—Rebecca West (1892–1983)
“Children should know there are limits to family finances or they will confuse “we can’t afford that” with “they don’t want me to have it.” The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.”
—Jean Ross Peterson (20th century)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)