Dear all,

 

I'm a math student and i facing problem to solve these question.hope all of u can help.

 

 

1. what would be wrong with defining matrix multiplication for matrices of the same size by multiplying them entry-by-entry, as with addition and subtraction?

 

2. discuss the relative merits of computing determinants between cofactor expansion method and row reduction method.

 

3. under what conditions will a matrix be diagonalizable? Are there any cases where one can tell at a glance whether a matrix is diagonalizable?

 

 

hope someone can help me...thanks a lot..

1. what would be wrong with defining matrix multiplication for matrices of the same size by multiplying them entry-by-entry, as with addition and subtraction?

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You can do that and it comes under the name of Hadamard or Schur product.

 

The biggest problem with it I can see is that it depends on a choice of basis for the underlying space.

 

2. discuss the relative merits of computing determinants between cofactor expansion method and row reduction method.

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Have a good think about this one.

 

3. under what conditions will a matrix be diagonalizable? Are there any cases where one can tell at a glance whether a matrix is diagonalizable?

 

Look at the rank of the matrix [math]P[/math] in [math]A = PDP^{-1}[/math] where [math]A[/math] is the matrix in question, [math]D[/math] is diagonal with the eigenvalues of [math]A[/math] and [math]P[/math] is invertible and consisting of the eigenvectors corresponding to the eigenvalues in [math]D[/math].

 

Two examples (and there are others) of invertible matrices are projections and symmetric matrices.

1. what would be wrong with defining matrix multiplication for matrices of the same size by multiplying them entry-by-entry, as with addition and subtraction?

You can do that and it comes under the name of Hadamard or Schur product.

 

The biggest problem with it I can see is that it depends on a choice of basis for the underlying space.

Surely, the most obvious problem is it not playing nice with the regular definition of matrix multiplication. Hadamard products aren't as generally useful as regular multiplication.

thanks for your post..^^..

i think it help me a lot.

but can you explain what is the meaning of "choice of basis for the underlying space"?can u give some example? i really not very understand what is the meaning.thanks again.

Think of matrices as tensors on space and then consider if the Hadamard product is also a genuine tensor. I have not worked it out, but I am sure the result is not really a matrix.

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Surely, the most obvious problem is it not playing nice with the regular definition of matrix multiplication. Hadamard products aren't as generally useful as regular multiplication.

 

I expect the reason for this is because of what I have said.