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I ran this and got some errors. Does anyone know how to solve this? First I want to caluclate A^2(A+I) with A = {{0, 1, 0}, {0, 0, 0}, {0, 0, -1}}.

MatrixFunction[#^2.(# + IdentityMatrix[3]) &, {{0, 1, 0}, {0, 0,
0}, {0, 0, -1}}]

StringForm::sfr: Item 1 requested in "Cannot compute the matrix function for the function 1." out of range; 0 items available. MatrixFunction::nosol: Cannot compute the matrix function for the function 1.

Second, I want to calculate A2^k:

A2= {{-0.5, 3}, {0, -0.8}};
MatrixFunction[#^k &, A2] // 
  FullSimplify[#, k > 0 && k \[Element] Integers] & // MatrixForm

MatrixFunction::fnnum: Unable to compute the matrix function because the function #1^k& at a numeric value is not numeric.

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  • 2
    $\begingroup$ I'm not sure what you're trying to do. $\endgroup$
    – chuy
    May 11, 2021 at 19:35
  • $\begingroup$ You can't have a matrix in the pure function in the first example. Use MatrixPower instead for the second example. $\endgroup$
    – Carl Woll
    May 11, 2021 at 20:37
  • $\begingroup$ @chuy I updated the post $\endgroup$
    – emnha
    May 12, 2021 at 5:28
  • $\begingroup$ @CarlWoll I want to make a pure function #^2(#+I) where I is the identiy matrix. So how can I make that? $\endgroup$
    – emnha
    May 12, 2021 at 5:29

1 Answer 1

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A = RandomReal[{0, 1}, {3, 3}];
MatrixFunction[#^2 (# + 1) &, A] == A . A . (A + IdentityMatrix[3])
(*    True    *)

You need to use the number 1, not the unit matrix, in the definition of the function for MatrixFunction. As the documentation states,

For convergent power series, MatrixFunction[f,m] effectively evaluates the power series for the function f with ordinary powers replaced by matrix powers.

For the second case, as @CarlWoll says, use MatrixPower:

A2 = {{-0.5, 3}, {0, -0.8}};
MatrixPower[A2, k]
(*    {{0. + 1. (-0.5)^k, -10. (-0.8)^k + 10. (-0.5)^k},
       {0.,               0. + 1. (-0.8)^k}}                *)

For some reason, MatrixFunction[#^k &, A2] does not seem to work if k is unspecified, even though the exact calculation works without a problem:

A2 = {{-1/2, 3}, {0, -4/5}};
MatrixFunction[#^k &, A2]
(*    {{(-1/2)^k, 5 (-1)^k 2^(1 - k) + (-1)^(1 + k) 2^(1 + 2 k) 5^(1 - k)},
       {0,        (-4/5)^k}}                                                   *)
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  • $\begingroup$ This would be complete if why ` MatrixFunction[#^k &, A2]` does not work is answered. $\endgroup$
    – emnha
    May 14, 2021 at 4:30
  • $\begingroup$ @anhnha see update $\endgroup$
    – Roman
    May 14, 2021 at 7:30
  • $\begingroup$ could you add the why the exact numbers make it work? $\endgroup$
    – emnha
    May 14, 2021 at 7:41
  • 1
    $\begingroup$ If I knew, I would have. $\endgroup$
    – Roman
    May 14, 2021 at 8:14

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