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Proof of the Singular Value Decomposition (SVD)

Title slide for the proof of the Singular Value Decomposition (SVD).

Theorem

The Singular Value Decomposition states that every real matrix can be factored into two orthogonal matrices and a rectangular diagonal matrix of singular values.

Handwritten proof slide stating the Singular Value Decomposition theorem for a real matrix.

Proof

Handwritten slide beginning the SVD proof and setting up the matrix and decomposition.

Show ATA is Symmetric

This symmetry connects the SVD to the Spectral Theorem, allowing the eigenstructure of ATA to guide the construction.

Handwritten proof slide showing that AᵀA is a real symmetric matrix.

Use the Spectral Theorem

Handwritten SVD proof slide applying the Spectral Theorem to the symmetric matrix AᵀA.

Form a Basis for ATA

Handwritten proof slide forming an orthonormal basis of eigenvectors for AᵀA.

Show the Eigenvalues are Nonnegative

Handwritten SVD proof showing why the eigenvalues of AᵀA are nonnegative, step 1.
Handwritten SVD proof showing why the eigenvalues of AᵀA are nonnegative, step 2.

Define the Singular Values and Left Singular Vectors of A

The eigenvalues and eigenvectors of ATA now provide the quantities used to construct the singular values and the corresponding left singular vectors.

Handwritten proof slide defining singular values from eigenvalues of AᵀA and constructing the left singular vectors.

Show that the Left Singular Vectors are Orthonormal

Handwritten SVD proof showing that the left singular vectors are orthonormal, step 1.
Handwritten SVD proof showing that the left singular vectors are orthonormal, step 2.
Handwritten SVD proof showing that the left singular vectors are orthonormal, step 3.

Determine the Number of Positive Singular Values of A

The rank identifies exactly how many nonzero eigenvalues of ATA contribute positive singular values to the decomposition.

Handwritten SVD proof using rank to determine the number of positive singular values, step 1.
Handwritten SVD proof using rank to determine the number of positive singular values, step 2.
Handwritten SVD proof using rank to determine the number of positive singular values, step 3.
Handwritten SVD proof using rank to determine the number of positive singular values, step 4.

Form the Left Singular Vector Matrix

Handwritten SVD proof slide forming the matrix U from the left singular vectors.

Extend the Orthogonal Set of Left Singular Vectors

Completing this orthonormal set provides the additional columns needed to build the full orthogonal matrix U.

Handwritten SVD proof extending the left singular vectors to an orthonormal basis of ℝⁿ, step 1.
Handwritten SVD proof extending the left singular vectors to an orthonormal basis of ℝⁿ, step 2.

Show U is Orthogonal

Handwritten SVD proof showing that the matrix U is orthogonal.

Construct the Singular Values Matrix

Handwritten SVD proof constructing the rectangular diagonal singular value matrix Σ.

Show AV = UΣ

Handwritten SVD proof establishing the matrix identity AV = UΣ, step 1.
Handwritten SVD proof establishing the matrix identity AV = UΣ, step 2.
Handwritten SVD proof establishing the matrix identity AV = UΣ, step 3.
Handwritten SVD proof establishing the matrix identity AV = UΣ, step 4.
Handwritten SVD proof establishing the matrix identity AV = UΣ, step 5.

Show A = UΣVT

These constructions bring the right singular vectors, left singular vectors, and singular values together in the factorization A = UΣVT.

Handwritten final SVD proof slide deriving A = UΣVᵀ.
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