Compute the square root of a symbolic matrix:

Verify that it is, indeed, a square root:

Note that this is different from simply computing the square root of each entry:

Compute the logarithm of a 3×3 matrix m:

Verify that the exponential of the result is the original matrix:

Compute a matrix polynomial, specifying the polynomial as a pure function:

Find the second inverse matrix power applied to a particular vector:

This is a more efficient way of computing :

Show that a matrix is a root of its characteristic polynomial:

The solution of , , for a scalar symbol is given with:

If is a matrix, the solution can be computed using matrix functions in the scalar solution:

Find the matrix that satisfies :

Confirm the formula for a Jordan matrix consisting of a single chain for the following matrix :

Using the function 1& returns an identity matrix:

Using the function 1/#& is the same as using Inverse:

Using a function that is a power is equivalent to using MatrixPower:

MatrixFunction[Exp,m] is essentially equivalent to MatrixExp[m]:

MatrixFunction effectively uses the power series, with Power replaced by MatrixPower:

Just as , MatrixExp[MatrixFunction[Log,m]] equals m:

If m is diagonal, MatrixFunction[f,m] merely applies f to each element of the diagonal of m:

If m is upper triangular, MatrixFunction[f,m] is also upper triangular:

The analogous statement holds for lower-triangular matrices:

If is diagonalizable with and the eigenvectors are well conditioned, then :

can be computed from the JordanDecomposition as :

Moreover, is zero except in upper-triangular blocks delineated by s in the superdiagonal:

A matrix is a root of its characteristic polynomial:

The scalar function can have symbolic derivatives for exact or symbolic matrices:

Compare with this example:

MatrixFunction does not work with non-differentiable functions such as Abs:

It will return a matrix function for an exact input matrix, but the result is meaningless because Abs does not have a first or second derivative:

MatrixFunction does not return a result when the scalar function or any of its initial derivatives are not defined at matrix eigenvalues:

The scalar function f has poles (singularities) at 1, 2, and 3:

If a scalar function is not analytic and a matrix eigenvalue is close to a function pole, the resulting matrix is usually incorrect: