Details and Options
- A matrix m is diagonalizable if it is similar to a diagonal matrix. That is, if there is a nonsingular matrix such that is diagonal.
- DiagonalizableMatrixQ works for symbolic as well as numerical matrices.
- For approximate matrices, the option Tolerance->t can be used to indicate that the determinant of eigenvectors v of the matrix m satisfying Abs[Det[v]]≤t is taken to be zero, which means that the eigenvectors v are linearly dependent and the matrix m is not diagonalizable.
- The option Tolerance has Automatic as its default value.
Examplesopen allclose all
Basic Examples (2)
Basic Uses (5)
Use DiagonalizableMatrixQ with an arbitrary-precision matrix:
Use DiagonalizableMatrixQ with a symbolic matrix:
Adjust the option Tolerance so that the linear dependence of the eigenvectors is detected:
If a matrix is diagonalizable, it can be decomposed as , where is nonsingular and is diagonal. This can be used to compute a matrix function by using its power series representation as , and can be computed by simply applying to each of the diagonal elements. Use this method to compute several different matrix functions:
Compute and confirm using MatrixPower:
Compute and confirm using MatrixExp:
Compute and confirm using MatrixLog:
Compute and confirm using MatrixFunction:
Verify the solution using DSolveValue:
Use the fact that the recurrence relation for Fibonacci, , can be written in matrix form as to derive a formula for the pair . The solution to the recurrence relation is , where :
Thus, the matrix power can be computed simply using Eigensystem[m]:
Confirm the result using Fibonacci:
Confirm using NormalMatrixQ:
A normal matrix like can be unitarily diagonalized using Eigensystem:
In quantum mechanics, the energy operator is called the Hamiltonian , and a state with energy evolves according to the Schrödinger equation . An important assumption is that any state can be written as a sum of eigenstates. Show that this is the case for Hamiltonian for a spin-1 particle in constant magnetic field in the direction:
Many matrix distributions produce matrices that are guaranteed to be diagonalizable, including CircularRealMatrixDistribution:
Properties & Relations (9)
A diagonalizable matrix can be factored as with diagonal using Eigensystem:
It only has two independent eigenvectors, as indicated by the zero vector returned by Eigensystem:
A matrix m is diagonalizable if and only if the matrix of JordanDecomposition[m] is diagonal:
Use JordanDecomposition to write as with diagonalizable and nilpotent:
Wolfram Research (2014), DiagonalizableMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/DiagonalizableMatrixQ.html.
Wolfram Language. 2014. "DiagonalizableMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DiagonalizableMatrixQ.html.
Wolfram Language. (2014). DiagonalizableMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiagonalizableMatrixQ.html