Section 5 — Q/X/XHat/P identities and approximations #
Late-stage algebraic identities and approximation lemmas for the paper's
Q/X/XHat/P intermediate layer.
Normalizing the images of positive Gram eigenvectors gives an orthonormal family.
This is the elementary singular-vector calculation underlying the rectangular
polar-decomposition route. If v_i are orthonormal eigenvectors of L†L with
positive eigenvalues λ_i, then the vectors λ_i^{-1/2} L v_i are
orthonormal in the target Hilbert space.
Matrix form of
orthonormal_normalized_image_of_adjoint_comp_eigenvectors.
For a rectangular matrix X, orthonormal eigenvectors of the Gram operator
Xᴴ * X with positive eigenvalues yield orthonormal normalized images under
X. This is the first local linear-algebra step toward constructing the
rectangular polar coisometry required for the QXP Xhat layer.
Spectral form of orthonormal_normalized_matrix_image_of_gram_eigenvectors.
After the right Gram matrix of X has been identified with a Hermitian
operator Q, its positive spectral subspace gives an orthonormal family of
normalized images under X. This is the form used when the rectangular polar
construction is indexed by the positive spectrum of the total Q operator.
The matrix whose rows are the normalized images of a prescribed Gram eigenvector family.
Equations
- MIPStarRE.LDT.MakingMeasurementsProjective.normalizedMatrixImageRows X v lam = Matrix.of fun (i : κ) (r : μ) => (↑(1 / √(lam i)) • (Matrix.toEuclideanLin X) (v i)).ofLp r
Instances For
The matrix of normalized images indexed by the positive spectrum of a Hermitian Gram operator.
Equations
- One or more equations did not get rendered due to their size.
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The normalized positive Gram images assemble into a coisometry matrix.
This is the row-matrix form of the preceding orthonormality statement. It is
the bridge from the singular-vector calculation to the matrix equation
Xhat Xhat† = I used by the QXP layer.
The transpose of the normalized-image row matrix satisfies the mixed Gram identity on the chosen eigenvector family.
This is the finite-dimensional calculation [ X^\dagger(\lambda_i^{-1/2}Xv_i)=\lambda_i^{1/2}v_i, ] recorded as a matrix identity with one column for each eigenvector.
Spectral form of normalized_matrix_image_rows_transpose_mixed.
On the positive spectral part of the Gram operator, the adjoint of X sends
the normalized image of an eigenvector back to the eigenvector multiplied by the
positive square root of its eigenvalue. This is the matrix identity which
records the nonzero singular values in the rectangular polar construction.
Spectral expansion of the CFC square root of a positive Hermitian matrix.
This is the square-root analogue of the Hermitian spectral expansion. It is
used below to compare the mixed product produced by the positive Gram-image
columns with the operator CFC.sqrt Q on the whole ambient space, including
the zero eigenspace.
Rows dual to the positive Gram eigenvectors.
Multiplying the positive mixed columns by this matrix sums those columns against the conjugate eigenvector coordinates. Thus the product is the spectral expansion of the square root of the Gram operator.
Equations
- MIPStarRE.LDT.MakingMeasurementsProjective.positiveGramSpectrumRightRows Q hQ = Matrix.of fun (i : { i : ι // 0 < hQ.eigenvalues i }) (j : ι) => star ((hQ.eigenvectorBasis ↑i).ofLp j)
Instances For
The conjugate eigenvector rows over the positive Gram spectrum are orthonormal.
This is the right-singular-vector companion to the normalized image-row
coisometry. It is independent of the matrix X; it uses only the
orthonormality of the Hermitian eigenvector basis for Q.
The positive Gram-image rows recover the square root of the Gram operator.
The identity
Xᴴ * (positiveGramSpectrumImageRows X Q hQ)ᵀ records the positive spectral
columns sqrt(λ_i) v_i. Multiplying by the conjugate eigenvector rows gives
the spectral expansion of CFC.sqrt Q; the zero eigenspace contributes nothing
because Q is positive semidefinite.
Zero Gram eigenvectors are killed by the rectangular matrix.
If Q = Xᴴ * X is positive semidefinite and an eigenvalue of Q is not
strictly positive, then it is zero. The corresponding eigenvector therefore
lies in the kernel of X. This is the algebraic reason why the arbitrary
completion directions in the rectangular polar construction do not contribute
to the mixed product Xᴴ * Xhat.
Vectors orthogonal to the positive Gram images are killed by X†.
The proof tests the adjoint vector against the complete eigenvector basis of
Q = Xᴴ * X. Positive eigenvectors are handled by the normalized-image
orthogonality assumption, while non-positive eigenvectors are killed by X
itself. This is the kernel statement needed when completing the positive
left singular-vector rows to a full unitary.
Spectral form of normalized_matrix_image_rows_mul_conjTranspose.
The rows indexed by the strictly positive eigenvalues of the Hermitian Gram operator are the normalized images of the corresponding Gram eigenvectors. They therefore assemble into a coisometry.
The strictly positive Gram spectrum has cardinality at most the row dimension of the rectangular matrix.
The proof uses the normalized-image orthonormal family above: an orthonormal
family in EuclideanSpace ℂ μ is linearly independent, so its index set cannot
be larger than the dimension of that space.