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haruspex said:What would a basis look like? It would be set of nxn matrices such that... you can do what with them?
Yes. What's the simplest matrix you can think of that might be useful in creating such a basis?Clandry said:For this case, a basis consists of all matrices such that all nxn diagonal matrices can be written as a linear combination of them?
haruspex said:Yes. What's the simplest matrix you can think of that might be useful in creating such a basis?
haruspex said:Ok. Now try 3x3.
You had 2 for 2x2 and 3 for 3x3. Why would you get infinitely many for nxn?Clandry said:if it's an nxn matrix, wouldn't that give an infinite amount of matrices for the bases?
haruspex said:You had 2 for 2x2 and 3 for 3x3. Why would you get infinitely many for nxn?
A diagonal matrix is a square matrix where all the elements outside the main diagonal are zero. The main diagonal contains the elements of the matrix, with all other elements being zero.
Finding the basis for a diagonal matrix is important because it allows us to represent any vector in the vector space as a linear combination of the basis vectors. This simplifies many calculations and makes it easier to understand the properties of the matrix.
To find the basis for a diagonal matrix, we can look at the non-zero elements on the main diagonal. These elements will form the basis vectors for the matrix.
Yes, a diagonal matrix can have multiple bases. This is because any non-zero scalar multiple of the basis vectors will still span the same vector space.
The basis for a diagonal matrix is directly related to its eigenvalues. The eigenvalues of a diagonal matrix correspond to the non-zero elements on the main diagonal, and the eigenvectors are the basis vectors for the matrix.