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December 31

Matrices and Indices

Can you raise a number to the power of a matrix and if so how?

By defining some arbitrary function on a number and a matrix and calling it a power. ...But more seriously, take a look at matrix exponential. I can think of at least one immediate generalization of this to arbitrary real or complex bases. « Aaron Rotenberg « Talk « 02:09, 31 December 2012 (UTC)[reply]

If a>0 and M is a square matrix, then a^M= e^{M log(a)}. If a is not a positive real then log(a) is multivalued. Bo Jacoby (talk) 05:30, 31 December 2012 (UTC).[reply]

So once you've worked out log(a) and multiplied it by M, how do you then raise e to the power of that? — Preceding unsigned comment added by 86.151.178.51 (talk) 14:25, 31 December 2012 (UTC)[reply]

Use

e^{M}=\sum _{n=0}^{\infty }{\frac {M^{n}}{n!}}

or

e^{M}=\lim _{n\rightarrow \infty }\left(1+{\frac {M}{n}}\right)^{n}

. Bo Jacoby (talk) 15:29, 31 December 2012 (UTC).[reply]

Bo, in the last expression, is "1" supposed to be the identity matrix "I"? Otherwise it's adding a scalar 1 to a matrix M/n. Is there a reference for this expression with the identity matrix?--if so, it's a nice intuitive generalization of the scalar formula, and I suggest you put it into the matrix exponential article. Duoduoduo (talk) 12:30, 2 January 2013 (UTC)[reply]

Mathematicians often use 1 to denote the identity matrix, or more generally the identity in any algebra over a field. There is no risk of confusion, since an algebra contains an isomorphic copy of its field of scalars, so it is natural to identify the scalars with multiples of the identity. This practice is very widespread. Sławomir Biały (talk) 13:12, 2 January 2013 (UTC)[reply]

Okay. Could someone provide a link to a citation for Bo's last formula above, so I can put it into the article? Duoduoduo (talk) 14:17, 2 January 2013 (UTC)[reply]

Exponential_function#Matrices_and_Banach_algebras. Bo Jacoby (talk) 09:26, 3 January 2013 (UTC).[reply]

Also, in the above-referenced article matrix exponential, see the sections "Computing the matrix exponential" and "Calculations". Duoduoduo (talk) 15:39, 31 December 2012 (UTC)[reply]