a. Consider the matrix product Q1=Q2 S, where both Q1 and Q2 are n ×m matrices with orthonormal

step 1 For this exercise A, we have that Q sub 1 equals Q sub 2 S. That S equals Q sub 2 transpose time

A. Consider the matrix product Q1=Q2 S, where both Q1 and Q2...

a. Consider the matrix product $Q_{1}=Q_{2} S$, where both $Q_{1}$ and $Q_{2}$ are $n \times m$ matrices with orthonormal columns. Show that $S$ is an orthogonal matrix. Hint: Compute $Q_{1}^{T} Q_{1}=\left(Q_{2} S\right)^{T} Q_{2} S$. Note that $Q_{1}^{T} Q_{1}=Q_{2}^{T} Q_{2}=I_{m}$ b. Show that the $Q R$ factorization of an $n \times m$ matrix $M$ is unique. Hint: If $M=Q_{1} R_{1}=Q_{2} R_{2}$ then $Q_{1}=Q_{2} R_{2} R_{1}^{-1}$. Now use part (a) and Exercise 50a.

a. Consider the matrix product $Q_{1}=Q_{2} S$, where both $Q_{1}$ and $Q_{2}$ are $n \times m$ matrices with orthonormal columns. Show that $S$ is an orthogonal matrix. Hint: Compute $Q_{1}^{T} Q_{1}=\left(Q_{2} S\right)^{T} Q_{2} S$. Note that $Q_{1}^{T} Q_{1}=Q_{2}^{T} Q_{2}=I_{m}$
b. Show that the $Q R$ factorization of an $n \times m$ matrix $M$ is unique. Hint: If $M=Q_{1} R_{1}=Q_{2} R_{2}$ then $Q_{1}=Q_{2} R_{2} R_{1}^{-1}$. Now use part (a) and Exercise 50a.

Key Concepts

Uniqueness in Matrix Factorization

The uniqueness of certain matrix factorizations, such as the QR factorization, relies on the specific structure imposed on the factors, like Q having orthonormal columns and R being upper triangular with positive diagonal entries. By comparing two different factorizations of the same matrix, one can show that the intervening matrix connecting the two must be orthogonal and, under further restrictions, equal to the identity. This ensures that the factorization is unique.

QR Factorization

The QR factorization is a method of decomposing a matrix into a product of two matrices: Q, which has orthonormal columns, and R, which is an upper triangular matrix. This factorization is very useful in numerical linear algebra for solving least squares problems and is foundational in many algorithms, including eigenvalue computations.

Orthonormal Columns

In a matrix, columns are called orthonormal if each column has unit norm and the columns are mutually orthogonal. This property implies that the transpose of the matrix multiplied by itself yields an identity matrix. Orthonormality is key in many matrix decompositions and ensures stability and simplicity in computations.

Orthogonal Matrices

An orthogonal matrix is a square matrix whose columns (and rows) are orthonormal. One of the defining properties of an orthogonal matrix is that its transpose is equal to its inverse, meaning S?S = SS? = I. This property is crucial when showing that a transformation preserves lengths and angles, and it is often used to prove uniqueness in matrix factorizations.

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