Equivalent Matrices: Definition & A=B

In summary, the definition of equivalent matrices is that two matrices, A and C, are considered equivalent if there exists an invertible matrix, B, that can be used to transform one matrix into the other. This can also be seen as both matrices representing the same linear transformation on different vector spaces, but in different bases.
  • #1
iVenky
212
12
What is the exact definition for equivalent matrices?

Is it necessary that it should be A = B if A,B are two matrices?

Thanks a lot.
 
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  • #2
The standard definition of "equivalent" for matrices (in any Linear Algebra text) is
"Matrices A and C are equivalent if and only if there exist an invertible matrix, B, such that BA= CB." Since B is invertible, that is the same as saying that [itex]A= B^{-1}CB[/itex] as well as [itex]C= BAB^{-1}[/itex]. From a more abstract point of view, matrices A and C are equivalent if and only if they represent the same linear transformation, on some vector spaces, as represented in different bases.
 

FAQ: Equivalent Matrices: Definition & A=B

What are equivalent matrices?

Equivalent matrices are matrices that have the same size and the same corresponding elements. This means that if two matrices, A and B, are equivalent, each element in A will have the same value as the corresponding element in B.

What is the definition of equivalent matrices?

The definition of equivalent matrices is two matrices, A and B, are considered equivalent if they have the same size and corresponding elements.

How can you determine if two matrices are equivalent?

To determine if two matrices are equivalent, you can compare the size of the matrices and check if each element in one matrix has the same value as the corresponding element in the other matrix.

What is the importance of equivalent matrices?

Equivalent matrices are important in understanding the properties and operations of matrices. They allow us to manipulate and solve equations involving matrices, as well as simplify complex matrix operations.

Can equivalent matrices be used in place of each other?

Yes, equivalent matrices can be used interchangeably in mathematical operations. This is because they have the same size and corresponding elements, so they will produce the same results when used in calculations.

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