# Solving Linear Algebra Problem: Eliminating p_2 with Diagonal D

• hotvette
In summary, the conversation discusses a system of equations involving matrices and unknown variables. The lower right submatrix is diagonal and therefore, p_2 can be easily eliminated from the system, resulting in a smaller system to solve for p_1 alone. The process involves using the inverse of the diagonal matrix D to solve for p_2 and then substituting the value of p_2 into the other equation to eliminate it.
hotvette
Homework Helper
I have a book that goes though a detailed development of the following:

$$\left[\begin{matrix}J^TJ & J^TV\\ VJ & D \end{matrix}\right]\left[\begin{matrix}p_1 \\ p_2\end{matrix}\right] = \left[\begin{matrix}J^Tr\\ Vr + Ds \end{matrix}\right]$$

where V and D are diagonal matrices and everything is known except $p_1$ and $p_2$.

Then it says "since the lower right submatrix D is diagonal, it is easy to eliminate $p_2$ from this system and obtain a smaller n x n system to be solved for $p_1$ alone." The implication is that it's so easy, explanation isn't needed. However, I don't see it.

Can someone explain how to eliminate $p_2$ given D is diagonal?

I've managed to come up with:

$$[VJ(J^TJ)^{-1}J^TV - D]p_2 = VJ(J^TJ)^{-1}J^Tr - Vr - Ds$$

but this seems far more complicated than what is implied and I don't see how D being diagonal simplifies anything.

Last edited:
If $D$ is diagonal with non zero eigenvalues, then it has an inverse $D^{-1}$. Thus from

$$V\,J\,p_1+D\,p_2=V\,r+D\,s$$

you can solve for $p_2$, i.e.

$$p_2=D^{-1}\,(V\,r+D\,s)-D^{-1}\,V\,J\,p_1$$

and use this to eliminate $p_2$ from the other equation.

Thanks. I can't believe I didn't see it. So simple.

## 1. How do you solve a linear algebra problem that involves eliminating p_2 with diagonal D?

To solve a linear algebra problem involving eliminating p_2 with diagonal D, you can use the Gauss-Jordan elimination method. This method involves transforming the given system of equations into an augmented matrix and then performing row operations to eliminate the variable p_2. The resulting matrix will have a row representing the solution for p_2, which can be used to solve for the other variables.

## 2. What is the purpose of eliminating p_2 with diagonal D in a linear algebra problem?

Eliminating p_2 with diagonal D is a common step in solving linear algebra problems because it simplifies the system of equations and makes it easier to solve for the remaining variables. It essentially reduces the number of unknowns in the system, making it more manageable to find a solution.

## 3. Can you provide an example of a linear algebra problem that involves eliminating p_2 with diagonal D?

One example of a linear algebra problem that involves eliminating p_2 with diagonal D is:
2p_1 + 3p_2 = 10
4p_1 - p_2 = 5
The diagonal D in this case would be the coefficient matrix on the left side, which is [2 3; 4 -1]. By performing row operations on this augmented matrix, we can eliminate p_2 and find its value.

## 4. Are there any special techniques or tricks for eliminating p_2 with diagonal D?

There are no special techniques or tricks for eliminating p_2 with diagonal D other than following the steps of the Gauss-Jordan elimination method. However, it is important to keep track of the row operations performed to ensure the resulting matrix is in the correct form for solving the remaining variables.

## 5. What are some common mistakes to avoid when eliminating p_2 with diagonal D in a linear algebra problem?

One common mistake to avoid when eliminating p_2 with diagonal D is forgetting to perform the same row operations on both sides of the augmented matrix. This can lead to an incorrect solution for p_2 and the remaining variables. It is also important to double-check the resulting matrix to ensure it is in the correct form for solving the remaining variables.

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