The .^ operator is generally the same as ^, except when dealing with matrices. While ^ does quite a lot of matrix-specific stuff (check out its page for more information), .^ just applies it element-by-element:
:[a,b;c,d] .^ 2 [a^2 b^2] [c^2 d^2]
The command can handle any choice of matrix and scalar as the base and exponent. However, if you're raising a constant number to a matrix power, be careful that the dot in .^ is not confused for a decimal point, by adding extra spaces:
:2.^[a,b;c,d] Error: Data type :2 .^ [a,b;c,d] [2^a 2^b] [2^c 2^d]
Although this doesn't come up often, be aware that .^, like ^, is evaluated from right to left: a.^b.^c is calculated as a.^(b.^c), not as (a.^b).^c.
240 - Dimension mismatch happens when a matrix is raised to the power of another matrix, with different dimensions.