Solve GAXPY Operations for Matrix A & Vector x

[lolittaFarhat]

Homework Statement

Let A be an nxn matrix belonging to R and x be a vector of length k belonging to R. Find the first column of
M = (A − x1I)(A − x2I)...(A − xrI) using a sequence of GAXPY’s operations.

Homework Equations

GAXPY: General matrix A multiplied by a vector X plus a vector Y

The Attempt at a Solution

I tried to figure it out by writing A and x explicitly and then multiplying (A-x1I) ...(A-xrI) but it was so messy and i did not get any result, i want a hint how to start the solution.

 

[Mark44]

Homework Statement

Let A be an nxn matrix belonging to R and x be a vector of length k belonging to R.

The matrix and vector don't "belong to" R. I think what you mean is that the entries in A and x are real numbers. The matrix would be an element of $\mathbb{R}^{n x n}$ and the vector would be (I think) an element of $\mathbb{R}^k$, unless by "length" you mean its magnitude.

Find the first column of
M = (A − x1I)(A − x2I)...(A − xrI) using a sequence of GAXPY’s operations.

What is r in the equation above? In other words, how many factors are there on the right?

Homework Equations

GAXPY: General matrix A multiplied by a vector X plus a vector Y

The Attempt at a Solution

I tried to figure it out by writing A and x explicitly and then multiplying (A-x1I) ...(A-xrI) but it was so messy and i did not get any result, i want a hint how to start the solution.

 

[lolittaFarhat]

what you thought is absolutely right, A is a matrix in R^(nxn) and x is an element of R^K . r is a real variable that is equal to k. Sorry for misstating the problem statement.

 

[Mark44]

Is there any relationship between k and n? Such as $k \le n$?

If not, are there an arbitrary number of factors in M? In what you wrote, there last component of x in the matrix product is x_r. The exact statement of the problem would be helpful.

I don't know if this is the best approach, but I would start with small matrices, say 2 x 2 or 3 x 3 matrix, to get an idea of how things work. I can't give any more advice until I understand more of the details of the problem.

 

[lolittaFarhat]

A is an nxn matrix, I is its identity and should be also an nxn matrix, k must be equal to n because we want A-x_rI . Here is the exact statement of the problem:
Let A ∈R^nxn, x ∈ R^k. Find the first column of M = (A − x₁I)(A − x₂I)...(A − x_kI) using a sequence of GAXPY’s operations.

 

[Mark44]

Try what I suggested at the end of post #4, with a 3 x 3 matrix A. That's what I would start with, and it might give you some insight into what happens for larger matrices.

$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \end{bmatrix}$$
$$x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$