Question

If \(A\) is \(m \times n\) of rank \(r,\) show that \(A\) can be factored as \(A=P Q\) where \(P\) is \(m \times r\) with \(r\) independent columns, and \(Q\) is \(r \times n\) with \(r\) independent rows. [Hint: Let \(U A V=\left[\begin{array}{cc}I_{r} & 0 \\ 0 & 0\end{array}\right]\) by Theorem 2.5.3, and write \(U^{-1}=\left[\begin{array}{ll}U_{1} & U_{2} \\ U_{3} & U_{4}\end{array}\right]\) and \(V^{-1}=\left[\begin{array}{ll}V_{1} & V_{2} \\\ V_{3} & V_{4}\end{array}\right]\) in block form, where \(U_{1}\) and \(V_{1}\) are \(\left.r \times r .\right]\)

Short answer

A can be factored as \(A = PQ\) where \(P = U_1\) and \(Q = V_1\).

Step-by-step solution

Understand the Transformation

The exercise suggests using Theorem 2.5.3 which transforms the matrix \(A\) into a block matrix \(UAV = \begin{bmatrix} I_r & 0 \ 0 & 0 \end{bmatrix}\). Here, \(U\) and \(V\) are invertible matrices that diagonalize \(A\). The rank \(r\) of the matrix is the number of leading 1s in the identity matrix \(I_r\).

Define the Inverse Block Matrices

Next, define the inverse matrices as block matrices: \(U^{-1} = \begin{bmatrix} U_1 & U_2 \ U_3 & U_4 \end{bmatrix}\) and \(V^{-1} = \begin{bmatrix} V_1 & V_2 \ V_3 & V_4 \end{bmatrix}\). Here, \(U_1\) and \(V_1\) are \(r \times r\) matrices assuming \(A\) has full row rank of \(r\).

Substitute and Decompose the Identity Transformation

Substitute the identity transformation into \(A = U^{-1} \begin{bmatrix} I_r & 0 \ 0 & 0 \end{bmatrix} V^{-1}\). This expression represents the factorization we're seeking, where multiplication by these block matrices constructs the original matrix \(A\) using its independent columns and rows.

Identify the Matrices P and Q

Recognizing the block forms, assign \(P = U_1\) (which is \(m \times r\)) and \(Q = V_1\) (which is \(r \times n\)). These matrices satisfy the requirement that \(P\) has \(r\) independent columns, and \(Q\) has \(r\) independent rows, as constrained by the identity matrix in the transformation.

Key concepts

Rank of a Matrix

The rank of a matrix is a fundamental concept in linear algebra. It tells us about the dimension of the vector space spanned by its rows or columns. When we say a matrix has a rank of \(r\), it means:

• There are \(r\) linearly independent rows.
• There are \(r\) linearly independent columns.
• The maximum number of linearly independent row vectors and column vectors is \(r\).

In essence, the rank provides insight into the complexity of a matrix—how much it can be 'stretched' without losing its distinctive properties. It is closely linked to two very important operations:

• Row reduction (forming a row echelon form)
• Column reduction (forming a column echelon form)

These processes help us determine linear dependencies among the vectors that the matrix represents. High rank indicates less dependency among rows or columns, whereas a lower rank suggests more dependency.

Independent Columns and Rows

The idea of independence is central to linear algebra. An independent set of columns (or rows) in a matrix means:

• None of the columns (or rows) can be written as a linear combination of the others.
• If one column (or row) changes, the overall span or the space they cover changes.

To better understand this, consider the matrix with just enough independent columns and rows to form basis vectors. It implies that these columns and rows fully utilize the given space, without redundancy. This independence is crucial for various operations:

• Determining solutions to linear equations.
• Understanding transformations represented by the matrix.

Independent columns are pivotal for decomposing a matrix into simpler forms, like the factorization \(A = PQ\). In such factorizations, certain arrays like \(P\) and \(Q\) are constructed using these independent sets to retain the properties of the original matrix.

Block Matrices

Block matrices are a technique where a matrix is divided into multiple blocks or sections. This method is extremely useful because:

• It simplifies complex matrix operations by operating on manageable blocks rather than the whole matrix.
• These blocks often represent subproblems or smaller portions of a larger system.

When working with large matrices, breaking down the matrix into smaller, easy-to-handle sections can be efficient. Notably, in the solution of our problem, the matrix \(A\) was transformed into a block matrix form using invertible matrices \(U\) and \(V\):\[UAV = \begin{bmatrix} I_r & 0 \ 0 & 0 \end{bmatrix}\]
Here, \(I_r\) represents the identity block that signifies the preservation of \(r\) independent dimensions. This block matrix facilitates the decomposition, allowing us to identify matrices like \(P\) and \(Q\) that encapsulate the independent row and column spaces.