Is EAE^(-1) = E^(-1)AE

Thread starter Woolyabyss
Start date Nov 21, 2014

In summary, the order of matrices is important in determining if E*A*E^(-1) = E^(-1)*A*E because matrix multiplication is not commutative. E represents the matrix used to transform A in a similarity transformation, and any type of square and invertible matrix can be used in this equation. To prove the equation, the definition of matrix multiplication and properties of invertible matrices can be used, along with the fact that the inverse of a matrix times itself is equal to the identity matrix. This equation is significant in linear algebra as it preserves certain properties of a matrix and allows for easier analysis and manipulation in various applications.

Nov 21, 2014

Woolyabyss

I know that matrix and its inverse are commutative i.e. E*E^(-1) = E^(-1)*E

but is a matrix and its inverse at either side of another matrix commutative?
E*A*E^(-1) = E^(-1)*A*E

Any help would be appreciated.

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Nov 21, 2014

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Insights Author

Not in general.
Otherwise every coordinate transformation would be equivalent to its inverse transformation.

1. Is the order of matrices important in determining if EAE^(-1) = E^(-1)AE?

Yes, the order of matrices is important in determining if E*A*E^(-1) = E^(-1)*A*E. This is because matrix multiplication is not commutative, meaning that the order in which matrices are multiplied can affect the result.

2. What does E represent in the equation EAE^(-1) = E^(-1)AE?

E represents the matrix that is being used to transform the matrix A. This transformation is known as a similarity transformation and is commonly used in linear algebra.

3. Can any type of matrix be used in this equation?

Yes, any type of matrix can be used in this equation as long as it is square and invertible. This means that the matrix must have the same number of rows and columns, and the determinant must not be equal to 0.

4. How do you prove that EAE^(-1) = E^(-1)AE?

To prove that E*A*E^(-1) = E^(-1)*A*E, you can use the definition of matrix multiplication and the properties of invertible matrices. You can also use the fact that the inverse of a matrix multiplied by the matrix itself is equal to the identity matrix.

5. What is the significance of EAE^(-1) = E^(-1)AE in linear algebra?

This equation is significant in linear algebra because it represents a similarity transformation, which preserves certain properties of a matrix such as eigenvalues and determinants. This allows for easier analysis and manipulation of matrices in various applications, such as solving systems of linear equations.