(a) According to Exercise 4.2.14, the Gram-Schmidt process...

(a) According to Exercise 4.2.14, the Gram-Schmidt process can also be applied to produce orthonormal bases of complex vector spaces. In the case of $\mathbb{C}^n$, explain how this is equivalent to the factorization of a nonsingular complex matrix $A=U R$ into the product of a unitary matrix $U$ (see Exercise 4.3.25) and a nonsingular upper triangular matrix $R$. (b) Factor the following complex matrices into unitary times upper triangular: (i) $\left(\begin{array}{rr}\mathrm{i} & 1 \\ -1 & 2 \mathrm{i}\end{array}\right)$, (ii) $$ \left(\begin{array}{cc} 1+i & 2-i \\ 1-i & -i \end{array}\right) $$ (iii) $\left(\begin{array}{lll}\mathrm{i} & 1 & 0 \\ 1 & \mathrm{i} & 1 \\ 0 & 1 & \mathrm{i}\end{array}\right)$, (iv) $$ \left(\begin{array}{ccc} \mathrm{i} & 1 & -\mathrm{i} \\ 1-\mathrm{i} & 0 & 1+\mathrm{i} \\ -1 & 2+3 \mathrm{i} & 1 \end{array}\right) $$ (c) What can you say about uniqueness of the factorization?

(a) According to Exercise 4.2.14, the Gram-Schmidt process can also be applied to produce orthonormal bases of complex vector spaces. In the case of $\mathbb{C}^n$, explain how this is equivalent to the factorization of a nonsingular complex matrix $A=U R$ into the product of a unitary matrix $U$ (see Exercise 4.3.25) and a nonsingular upper triangular matrix $R$.
(b) Factor the following complex matrices into unitary times upper triangular:
(i) $\left(\begin{array}{rr}\mathrm{i} & 1 \\ -1 & 2 \mathrm{i}\end{array}\right)$,
(ii)
$$
\left(\begin{array}{cc}
1+i & 2-i \\
1-i & -i
\end{array}\right)
$$
(iii) $\left(\begin{array}{lll}\mathrm{i} & 1 & 0 \\ 1 & \mathrm{i} & 1 \\ 0 & 1 & \mathrm{i}\end{array}\right)$,
(iv)
$$
\left(\begin{array}{ccc}
\mathrm{i} & 1 & -\mathrm{i} \\
1-\mathrm{i} & 0 & 1+\mathrm{i} \\
-1 & 2+3 \mathrm{i} & 1
\end{array}\right)
$$
(c) What can you say about uniqueness of the factorization?

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is one where all entries below the main diagonal are zero. In the QR factorization of a complex matrix, the R factor is an upper triangular matrix. This structure simplifies many computations including solving linear systems and performing further factorization or eigenvalue analysis.

Unitary Matrix

A unitary matrix in a complex vector space is the counterpart of an orthogonal matrix in real vector spaces. It is defined by the property that its conjugate transpose is equal to its inverse (U*U = I). Unitary matrices preserve the inner product, meaning they maintain the length of vectors and the angles between them, making them essential in preserving geometric properties during transformations and crucial in quantum mechanics and signal processing.

Uniqueness of Factorization

The QR factorization is unique up to the phases (complex signs) of the diagonal elements of the R matrix. In other words, if additional constraints are imposed such as requiring the diagonal entries of R to be positive (or to have a prescribed phase), then the factorization can be made unique. Without such conditions, different factorization choices can lead to equivalent decompositions.

Gram-Schmidt Process

This is an algorithm for converting a set of linearly independent vectors into a set of orthonormal vectors, preserving the span of the original set. In the context of complex vector spaces, the process is adapted by using the appropriate complex inner product, which results in an orthonormal basis composed of complex vectors. This orthonormal basis is fundamental for many applications including simplifying matrix representations and decompositions.

QR Factorization

QR factorization is a method of decomposing a nonsingular matrix into the product of a unitary (or orthogonal in the real case) matrix Q and an upper triangular matrix R. This process in complex vector spaces is essentially carried out by the Gram-Schmidt process where the matrix whose columns are the vectors being processed becomes Q, and the resulting coefficients from the process form the upper triangular matrix R. This factorization plays a central role in numerical linear algebra and solving least squares problems.

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