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Computing the k th power of a matrix needs k – 1 times the time of a single matrix multiplication, if it is done with the trivial algorithm (repeated multiplication). As this may be very time consuming, one generally prefers using exponentiation by squaring, which requires less than 2 log 2 k matrix multiplications, and is therefore much more ...
Then the behavior of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one impedance element (h 12), one admittance element (h 21), and two dimensionless elements (h 11 and h 22). Calculating a circuit now reduces to multiplying matrices.
For diagonalizable matrices, as illustrated above, e.g. in the 2×2 case, Sylvester's formula yields exp(tA) = B α exp(tα) + B β exp(tβ), where the B s are the Frobenius covariants of A. It is easiest, however, to simply solve for these B s directly, by evaluating this expression and its first derivative at t = 0 , in terms of A and I , to ...
In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication.It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although the naive algorithm is often better for smaller matrices.
The lower bound of multiplications needed is 2mn+2n−m−2 (multiplication of n×m-matrices with m×n-matrices using the substitution method, m⩾n⩾3), which means n=3 case requires at least 19 multiplications and n=4 at least 34. [40] For n=2 optimal 7 multiplications 15 additions are minimal, compared to only 4 additions for 8 multiplications.
The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by () = ().The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix itself, the result is the zero matrix: () =.
[citation needed] According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator T 1/2 such that T 1/2 is itself positive and (T 1/2) 2 = T. The operator T 1/2 is the unique non-negative square root of T. [citation needed] A bounded non-negative operator on a complex Hilbert space is self adjoint by ...
Given a formal Laurent series = =, the corresponding Hankel operator is defined as [2]: [] [[]]. This takes a polynomial [] and sends it to the product , but discards all powers of with a non-negative exponent, so as to give an element in [[]], the formal power series with strictly negative exponents.