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The calculation of the sequence , …, is known as Gram–Schmidt orthogonalization, and the calculation of the sequence , …, is known as Gram–Schmidt orthonormalization. To check that these formulas yield an orthogonal sequence, first compute u 1 , u 2 {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle } by substituting the ...
The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep ...
Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits an orthonormal basis; [7] furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension ...
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space). [4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with ...
In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information theory , for example in entanglement characterization and in state purification , and plasticity .
In matrix notation, = /, where has orthonormal basis vectors {} and the matrix is composed of the given column vectors {}. The matrix G − 1 / 2 {\displaystyle G^{-1/2}} is guaranteed to exist. Indeed, G {\displaystyle G} is Hermitian, and so can be decomposed as G = U D U † {\displaystyle G=UDU^{\dagger }} with U {\displaystyle U} a unitary ...
If the n × n matrix M is nonsingular, its columns are linearly independent vectors; thus the Gram–Schmidt process can adjust them to be an orthonormal basis. Stated in terms of numerical linear algebra, we convert M to an orthogonal matrix, Q, using QR decomposition.
The geometric content of the SVD theorem can thus be summarized as follows: for every linear map : one can find orthonormal bases of and such that maps the -th basis vector of to a non-negative multiple of the -th basis vector of , and sends the leftover basis vectors to zero.