Several matrix inverse properties

In summary, the conversation discusses computing the inverse of (VΔYT), where V is orthogonal, Δ is diagonal, and Y is nonsingular. It is mentioned that in general, (AB)-1 = B-1A-1. The question is raised about how to compute the inverse for multiple matrices and if the structure of the matrices (orthogonal, nonsingular, etc) affects the process. The expert then provides the solution, (ABC)-1 = C-1B-1A-1. The asker confirms this is what they were looking for.
  • #1
pob1212
21
0
Hi,

I'm specifically trying to compute (VΔYT)-1, where V is nxn and orthogonal, Δ is diagonal, and Y is nonsingular.

In general we have (AB)-1 = B-1A-1

But how do we do this in general for many matrices? Is there a method, and as long as the matrix dimensions agreed, does the structure (i.e. orthogonal, nonsingular, SPD, etc) matter when computing the inverse of multiple matrices?

I can't find this anywhere on the internet.

Thanks
 
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  • #2
[tex](ABC)^{-1}=(A(BC))^{-1}=(BC)^{-1}A^{-1}=C^{-1}B^{-1}A^{-1}[/tex]

Was that what you were looking for?
 
  • #3
exactly what I was looking for. Thanks
 

Related to Several matrix inverse properties

1. What is the definition of a matrix inverse?

A matrix inverse is a matrix that, when multiplied with the original matrix, produces the identity matrix. In other words, it undoes the effects of the original matrix.

2. How is the matrix inverse calculated?

The matrix inverse is calculated by using the formula (1/det(A)) * adj(A), where det(A) is the determinant of the original matrix and adj(A) is the adjugate matrix of the original matrix.

3. Can all matrices have an inverse?

No, not all matrices have an inverse. Only square matrices (matrices with the same number of rows and columns) and non-singular matrices (matrices with a non-zero determinant) have an inverse.

4. What are some properties of the matrix inverse?

Some properties of the matrix inverse include: the inverse of the inverse of a matrix is the original matrix, the inverse of a product of matrices is the product of the inverses of those matrices in reverse order, and the inverse of a transpose of a matrix is the transpose of the inverse of that matrix.

5. How is the matrix inverse used in practical applications?

The matrix inverse is used to solve systems of linear equations, find the coefficients in polynomial interpolation, and perform transformations in computer graphics. It is also used in data compression and error correction algorithms.

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