I am reading a proof of Schur's Theorem (statement of theorem and portion of proof up to where I have a question will be provided below) and am confused on the following:
Consider a $(k \times 1) \times (k \times 1)$ matrix, $A$. Let $\lambda_1$ be an eigenvalue of $A$, and let $w_1$ be a unit eigenvector belonging to $\lambda_1$. Using the Gram-Schmidt process, construct $w_2,\dots,w_{k+1}$ such that $\{w_1,w_2,\dots,w_{k+1}\}$ is an orthonormal basis for $\mathbb{C}^{k+1}$
I am confused on how the Gram-Schmidt process would be used to find an orthonormal basis in this case, since I thought that the Gram-Schmidt process is used to go from a basis to an orthonormal basis. Here we do not have a basis to "start with", so how would the Gram-Schmidt process be used?
To elaborate a bit, I was thinking that making the column vectors of the matrix $A$ perhaps form a basis, but without any assumptions these column vectors could be linearly dependent.
To clarify: I'm not looking for someone to give me the method of construct basis (that should just be Gram-Schmidt). Im asking how we can use Gram-Schmidt without starting with a basis (or, if we are starting with a basis, what is that basis in this case)
Theorem: For each $n\times n$ matrix $A$, there exists a unitary matrix $U$ such that $U^HAU$ is upper triangular.
(part of) Proof:
The proof is by induction on $n$. when $n=1$ the result is obvious. Assume the result holds for $k\times k$ matrices.
Let $A$ be a $(k\times 1)\times (k\times 1)$ matrix. Let $\lambda_1$ be an eigenvalue of $A$, and let $w_1$ be a unit eigenvector belonging to $\lambda_1$. Using the Gram-Schmidt process, construct $w_2,\dots,w_{k+1}$ such that $\{w_1,w_2,\dots,w_{k+1}\}$ is an orthonormal basis for $\mathbb{C}^{k+1}$
Best Answer
You can start the Gram-Schmidt process with an arbitrary basis whose first element is $w_1$. It's easy to see that such a thing exists. You can construct it as follows. At each step you'll have a set of linearly independent vectors. As long as there are fewer of them than the dimension of your space, take any vector that is not in the linear span of your set, and insert it into the set.