Advanced Linear Algebra: Foundations to Frontiers

In the below proof, it is really imporant to keep track of when a line is part of the partitioning of a matrix or vector, and when it denotes scalar division.

Choose \(\sigma_1 \) and \(\widetilde v_1 \in \C^n \) such that

• \(\| \widetilde v_1 \|_2 = 1 \text{;}\) and

• \(\sigma_1 = \| A \widetilde v_1 \|_2 = \| A \|_2 \text{.}\)

In other words, \(\widetilde v_1 \) is the vector that maximizes \(\max_{\| x \|_2 = 1} \| A x \|_2 \text{.}\)

Let \(\widetilde u_1 = A \widetilde v_1 / \sigma_1 \text{.}\) Then

Choose \(\widetilde U_2 \in \C^{m \times (m-1)} \) and \(\widetilde V_2 \in \C^{n \times (n-1)} \) so that

are unitary. Then

where \(w = \widetilde V_2^H A^H \widetilde u_1 \) and \(B = \widetilde U_2^H A \widetilde V_2 \text{.}\)

We will now argue that \(w = 0 \text{,}\) the zero vector of appropriate size:

Thus \(\sigma_1^2 \geq \sigma_1^2 + w^H w \) which means that \(w = 0 \) (the zero vector) and \(\widetilde U^H A \widetilde V = \left( \begin{array}{c | c} \sigma_1 \amp 0 \\ \hline 0 \amp B \end{array} \right)\) so that \(A = \widetilde U \left( \begin{array}{c | c} \sigma_1 \amp 0 \\ \hline 0 \amp B \end{array} \right) \widetilde V^H \text{.}\)

We will prove this for \(m \geq n \text{,}\) leaving the case where \(m \leq n \) as an exercise.

Proof by induction: Since \(m \geq n \text{,}\) we select \(m \) to be arbritary and induct on \(n \text{.}\)

• Base case: \(n = 1 \text{.}\)

  In this case \(A = \left( \begin{array}{c} a_1 \end{array} \right) \) where \(a_1 \in \Cm \) is its only column.

  Case 1: \(a_1 = 0 \) (the zero vector).

  Then

  \begin{equation*} A = \left( \begin{array}{c} 0 \end{array} \right) = \begin{array}[t]{c} \underbrace{ I_{m \times m} } \\ U \end{array} \left( \begin{array}{ c | c } ~ \amp ~ \\ \hline ~ \amp 0 \end{array} \right) \begin{array}[t]{c} \underbrace{ I_{1 \times 1} } \\ V^H \end{array} \end{equation*}

  so that \(U = I_{m \times m} \text{,}\) \(V = I_{1 \times 1} \text{,}\) and \(\Sigma_{TL} \) is an empty matrix.

  Case 2: \(a_1 \neq 0 \text{.}\)

  Then

  \begin{equation*} A = \left( \begin{array}{c} a_1 \end{array} \right) = \left( \begin{array}{c} u_1 \end{array} \right) \left( \| a_1 \|_2 \right) \end{equation*}

  where \(u_1 = a_1 / \| a_1 \|_2 \text{.}\) Choose \(U_2 \in \C^{m \times (m-1)} \) so that \(U = \left( \begin{array}{c | c} u_1 \amp U_2 \end{array} \right)\) is unitary. Then

  \begin{equation*} \begin{array}{rcl} A \amp = \amp \left( \begin{array}{c} a_1 \end{array} \right) \\ \amp = \amp \left( \begin{array}{c} u_1 \end{array} \right) ( \| a_1 \|_2 ) \\ \amp = \amp \left( \begin{array}{c| c} u_1 \amp U_2 \end{array} \right) \left( \begin{array}{c | c } \| a_1 \|_2 \amp ~\\ \hline 0 \amp ~ \end{array} \right) \left( \begin{array}{c} 1 \end{array} \right)^H \\ \amp = \amp U \Sigma V^H, \end{array} \end{equation*}

  where

  • \(U = \left( \begin{array}{c| c} u_1 \amp U_2 \end{array} \right) \text{,}\)

  • \(\Sigma = \left( \begin{array}{ c | c} \Sigma_{TL} \amp ~ \\ \hline 0 \amp ~ \end{array} \right) \) with \(\Sigma_{TL} = \left( \begin{array}{c} \sigma_1 \end{array} \right) \) and \(\sigma_1 = \| a_1 \|_2 = \| A \|_2 \)

  • \(V = \left( \begin{array}{c} 1 \end{array} \right) \text{.}\)

• Inductive step:

  Assume the result is true for matrices with \(1 \leq k \) columns. Show that it is true for matrices with \(k+1 \) columns.

  Let \(A \in \C^{m \times (k+1)} \) with \(1 \leq k \lt n \text{.}\)

  Case 1: \(A = 0 \) (the zero matrix)

  Then

  \begin{equation*} A = I_{m \times m} \left( \begin{array}{c |c } ~ \amp ~ \\ \hline 0_{m \times (k+1)} \amp ~ \end{array} \right) I_{(k+1) \times (k+1)} \end{equation*}

  so that \(U = I_{m \times m} \text{,}\) \(V = I_{(k+1) \times (k+1)} \text{,}\) and \(\Sigma_{TL} \) is an empty matrix.

  Case 2: \(A \neq 0 \text{.}\)

  Then \(\| A \|_2 \neq 0 \text{.}\) By Lemma 2.3.1.3, we know that there exist unitary \(\widetilde U \in \C^{m \times m} \) and \(\widetilde V \in \C^{(k+1) \times (k+1)} \) such that \(A = \widetilde U \left( \begin{array}{c|c} \sigma_1 \amp 0 \\ \hline 0 \amp B \end{array} \right)\widetilde V^H \) with \(\sigma_1 = \| A \|_2 \text{.}\)

  By the inductive hypothesis, there exist unitary \(\check U_B \in \C^{ (m-1) \times (m-1)} \text{,}\) unitary \(\check V_B \in \C^{k \times k} \text{,}\) and \(\check \Sigma_B \in \R^{(m-1) \times k}\) such that \(B = \check U_B \check \Sigma_B \check V_B^H \) where \(\check \Sigma_B = \FlaTwoByTwo{ \check \Sigma_{TL} }{ 0 }{ 0 }{ 0 } \text{,}\) \(\check \Sigma_{TL} = \diag{ \sigma_2, \cdots , \sigma_{r-1} } \text{,}\) and \(\sigma_2 \geq \cdots \geq \sigma_{r-1} \gt 0 \text{.}\)

  Now, let

  \begin{equation*} U = \widetilde U \left( \begin{array}{c | c} 1 \amp 0 \\ \hline 0 \amp \check U_B \end{array} \right), V = \widetilde V \left( \begin{array}{c | c} 1 \amp 0 \\ \hline 0 \amp \check V_B \end{array} \right), \mbox{ and } \Sigma = \left( \begin{array}{c | c} \sigma_1 \amp 0 \\ \hline 0 \amp \check \Sigma_B \end{array} \right). \end{equation*}

  (There are some really tough to see "checks" in the definition of \(U \text{,}\) \(V \text{,}\) and \(\Sigma \text{!!}\)) Then \(A = U \Sigma V^H \) where \(U \text{,}\) \(V \text{,}\) and \(\Sigma\) have the desired properties. Key here is that \(\sigma_1 = \| A \|_2 \geq \| B \|_2 \) which means that \(\sigma_1 \geq \sigma_2 \text{.}\)

• By the Principle of Mathematical Induction the result holds for all matrices \(A \in \C^{m \times n} \) with \(m \geq n \text{.}\)