Suppose A is an m ×n matrix with rank A=n. (a) Show that applying the GramSchmidt algorithm to t

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Suppose A is an m ×n matrix with rank A=n. (a) Show that...

Suppose $A$ is an $m \times n$ matrix with $\operatorname{rank} A=n$. (a) Show that applying the GramSchmidt algorithm to the columns of $A$ produces an orthonormal basis for img $A$. (b) Prove that this is equivalent to the matrix factorization $A=Q R$, where $Q$ is an $m \times n$ matrix with orthonormal columns, while $R$ is a nonsingular $n \times n$ upper triangular matrix. (c) Show that the $Q R$ program in the text also works for rectangular, $m \times n$, matrices as stated, the only modification being that the row indices $i$ run from 1 to $m$. (d) Apply this method to factor (i) $\left(\begin{array}{rr}1 & -1 \\ 2 & 3 \\ 0 & 2\end{array}\right)$, (ii) $\left(\begin{array}{rr}-3 & 2 \\ 1 & -1 \\ 4 & 1\end{array}\right)$, (iii) $\left(\begin{array}{rr}-1 & 1 \\ 1 & -2 \\ -1 & -3 \\ 0 & 5\end{array}\right)$ (iv) $$ \left(\begin{array}{rrr} 0 & 1 & 2 \\ -3 & 1 & -1 \\ -1 & 0 & -2 \\ 1 & 1 & -2 \end{array}\right) $$ (e) Explain what happens if $\operatorname{rank} A<n$.

Suppose $A$ is an $m \times n$ matrix with $\operatorname{rank} A=n$. (a) Show that applying the GramSchmidt algorithm to the columns of $A$ produces an orthonormal basis for img $A$. (b) Prove that this is equivalent to the matrix factorization $A=Q R$, where $Q$ is an $m \times n$ matrix with orthonormal columns, while $R$ is a nonsingular $n \times n$ upper triangular matrix. (c) Show that the $Q R$ program in the text also works for rectangular, $m \times n$, matrices as stated, the only modification being that the row indices $i$ run from 1 to $m$. (d) Apply this method to factor
(i) $\left(\begin{array}{rr}1 & -1 \\ 2 & 3 \\ 0 & 2\end{array}\right)$,
(ii) $\left(\begin{array}{rr}-3 & 2 \\ 1 & -1 \\ 4 & 1\end{array}\right)$,
(iii)
$\left(\begin{array}{rr}-1 & 1 \\ 1 & -2 \\ -1 & -3 \\ 0 & 5\end{array}\right)$
(iv)
$$
\left(\begin{array}{rrr}
0 & 1 & 2 \\
-3 & 1 & -1 \\
-1 & 0 & -2 \\
1 & 1 & -2
\end{array}\right)
$$
(e) Explain what happens if $\operatorname{rank} A<n$.

Key Concepts

Gram-Schmidt Process

The Gram-Schmidt process is an algorithm used to convert a set of linearly independent vectors into a set of orthonormal vectors that span the same subspace. This procedure iteratively subtracts out the components of each vector in the direction of the already established orthonormal vectors and then normalizes the result, ensuring that the new vectors are both orthogonal to each other and of unit length.

QR Factorization

QR factorization is a method of decomposing a matrix into a product of an orthonormal matrix Q and an upper triangular matrix R. This factorization is particularly important in numerical linear algebra because it simplifies the solution of systems of linear equations and least squares problems. It critically relies on the linear independence of the matrix's columns, which is assured when the matrix has full column rank.

Orthonormal Basis

An orthonormal basis of a subspace consists of vectors that are mutually perpendicular (orthogonal) and each of unit length (normalized). The advantage of using an orthonormal basis is that it simplifies many mathematical operations, such as projections and coordinate transformations, due to the inner product of any two distinct basis vectors being zero and the inner product of a basis vector with itself being one.

Matrix Rank

The rank of a matrix is a measure of the maximum number of linearly independent columns (or rows) it contains. A matrix having full column rank implies that its columns are linearly independent, a necessary condition for the successful application of the Gram-Schmidt process and the existence of a nonsingular factor in the QR factorization. If the rank is lower than the number of columns, the matrix is rank-deficient, which can lead to a failure in obtaining a complete set of orthonormal vectors or a invertible triangular factor.

Rectangular Matrices

Rectangular matrices, which have more rows than columns or vice versa, require a careful adaptation of algorithms like the Gram-Schmidt process and QR factorization. In these cases, the method must correctly account for the differing dimensions, ensuring that the computations of vector projections and orthogonalizations are performed over the correct range of indices. This often means iterating over the appropriate row or column indices to construct the appropriate orthonormal basis for the subspace spanned by the matrix's columns.

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