Help with simple matrix algebra

In summary, the conversation discusses a matrix problem where the goal is to find the matrix r. However, the given matrix H is not square and cannot be inverted, so the pseudoinverse method is used. It is mentioned that there are infinitely many solutions due to the full rank of the matrix A. A formula for the Moore-Penrose pseudoinverse is provided, but it is noted that using it may not lead to desired results due to the singular value decomposition of A.
  • #1
rusty009
69
0
Hi all,

I'm having trouble solving this matrix problem, basically I have,

s=r*H

where s = [ 1 1 0] and,

H=
1 0 1 1 0 0
0 1 1 0 1 0
1 1 0 0 0 1

I am trying to find out what the matrix r is but it won't work in matlab. I have tried inverting H but it isn't a square matrix. So I tried pseudoinverse psinv and it still doesn't seem to be working. Any help would be greatly appreciated.
 
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  • #2
Your problem is that this is impossible. H has three rows and 6 columns so r must have three columns. If it also has n rows, s would have to have n rows and 6 columns. Since s has one row, n can be 1 but s has 3 columns, not 6.
 
  • #3
Your equation is of the form :
[tex]
\vec{a} = \vec{b}A
[/tex]

Where b is the 1x3 vector we are looking for and a is a 1x6 vector . As you mentioned, your matrix isn't square an thus not normally invertible. By the Moore-Penrose Pseudoinverse, there does exist a solution (infinitely many actually).
Your matrix A is an mxn matrix where m<n and is also full-rank, so:
[tex]
A^\dagger = (A^TA)^{-1}A^T
[/tex]
Where A-dagger is the Moore-Penrose Pseudoinverse. Notice that only because A was full rank could you use this formula, otherwise you have to use the singular value decomposition .
I've tried [tex] A^\dagger = (A^TA)^{-1}A^T [/tex] with your matrix but [tex] A^TA [/tex] turns out to be singular. I suppose you could try to take the pseudoinverse of that too, but then I'm not sure if everything will work out as you had wanted.

Edit: I made a silly mistake with the size of r :redface: (as HallsofIvy pointed out). The above would have been useful for consistent sizes and a full rank matrix.
 
Last edited:

1. What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is used to represent and manipulate data in many areas of mathematics, statistics, and science.

2. What is meant by simple matrix algebra?

Simple matrix algebra refers to the basic operations and rules used to perform calculations with matrices, such as addition, subtraction, multiplication, and inversion. It does not involve advanced concepts like determinants or eigenvalues.

3. How do I add or subtract matrices?

To add or subtract matrices, they must have the same dimensions (same number of rows and columns). Simply add or subtract the corresponding elements in each matrix to get the result. For example, to add two 2x2 matrices A and B, add A11 + B11 for the first element of the result, A12 + B12 for the second element, and so on.

4. How do I multiply matrices?

To multiply matrices, the number of columns in the first matrix must match the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. Use the row-by-column method, where each element in the resulting matrix is calculated by multiplying the corresponding row of the first matrix by the corresponding column of the second matrix and adding the products.

5. How do I find the inverse of a matrix?

The inverse of a matrix A, denoted as A-1, is a matrix that, when multiplied by A, gives the identity matrix (a square matrix with 1s on the main diagonal and 0s everywhere else). To find the inverse of a matrix, use the Gauss-Jordan elimination method or the adjugate matrix method. It is important to note that not all matrices have an inverse, and square matrices with a determinant of 0 do not have an inverse.

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