To intuitively understand matrix exponents of the form e^(At), where A is a matrix, it is helpful to consider the analogy between matrix exponentiation and scalar exponentiation.
When we exponentiate a scalar, say e^x, we are essentially performing repeated multiplication of the base value (e) by itself, x number of times. The resulting value represents the exponential growth or decay over time.
In the case of matrix exponentiation, e^(At), the exponential function is applied to a matrix A multiplied by a scalar t. The matrix A represents the rate of change or transformation, and the scalar t represents time. By exponentiating the matrix A, we are essentially "multiplying" the matrix by itself, t number of times.
The exponential of a matrix can be computed using its power series expansion:
e^(At) = I + At + (A^2)t^2/2! + (A^3)t^3/3! + ...
Here, I is the identity matrix and A^n represents the matrix raised to the power of n.
Intuitively, the matrix exponential e^(At) represents a transformation that occurs over time, governed by the matrix A. Just as scalar exponentiation captures exponential growth or decay, matrix exponentiation captures how a matrix evolves or transforms over time.
The matrix A determines the nature of this transformation. For example, if A represents a diagonal matrix with eigenvalues λ_i on the diagonal, e^(At) will also be a diagonal matrix with elements e^(λ_i*t) on the diagonal. This indicates that each component of a vector will independently undergo exponential growth or decay based on the corresponding eigenvalue.
In summary, matrix exponentiation e^(At) represents the transformation of a matrix over time, analogous to scalar exponentiation representing exponential growth or decay. The matrix A determines the nature of the transformation, and the exponent t determines the duration or "power" of the transformation.