Bill Simpson said:
In Mathematica, matrices can be created by using the command "Array" or by typing the elements directly into a list and using the "MatrixForm" command to display it. For example, to create a 3x3 matrix with the elements 1, 2, 3, 4, 5, 6, 7, 8, 9, you can use either of the following commands:
MatrixForm[Array[{1, 2, 3, 4, 5, 6, 7, 8, 9}, {3, 3}]] or MatrixForm[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}]
To put a matrix in block diagonal form, you can use the "JordanDecomposition" command in Mathematica. This command will return a list of matrices that represent the block diagonal form of the original matrix. For example, if you have a matrix A, you can use the command "JordanDecomposition[A]" to obtain the block diagonal form of A.
Yes, you can also put a matrix in block diagonal form by using the "DiagonalMatrix" command. This command creates a diagonal matrix from a list of elements. To put a matrix A in block diagonal form, you can use the command "DiagonalMatrix[Eigenvalues[A]]". This will create a diagonal matrix with the eigenvalues of A on the diagonal, which will result in the block diagonal form of A.
Yes, you can use the "MatrixPlot" command in Mathematica to visualize the block diagonal form of a matrix. This command creates a plot of the matrix elements, with different colors representing different values. To use this command, you can simply input the block diagonal matrix obtained from the "JordanDecomposition" or "DiagonalMatrix" commands.
To convert a block diagonal matrix back to its original form, you can use the "KroneckerProduct" command in Mathematica. This command computes the Kronecker product of two matrices. To convert a block diagonal matrix A to its original form, you can use the command "KroneckerProduct[P, Q]" where P is the matrix obtained from the "JordanDecomposition" or "DiagonalMatrix" commands, and Q is the inverse of P. This will result in the original matrix A.