Search results
Results from the WOW.Com Content Network
Similarity is an equivalence relation on the space of square matrices. Because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator: Rank; Characteristic polynomial, and attributes that can be derived from it:
In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if = for some invertible n-by-n matrix P and some invertible m-by-m matrix Q.Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.
Another way to find the square root of an n × n matrix A is the Denman–Beavers square root iteration. [ 8 ] Let Y 0 = A and Z 0 = I , where I is the n × n identity matrix .
A square matrix that does not have a complete basis of eigenvectors, and is thus not diagonalizable. Derogatory matrix: A square matrix whose minimal polynomial is of order less than n. Equivalently, at least one of its eigenvalues has at least two Jordan blocks. [3] Diagonalizable matrix: A square matrix similar to a diagonal matrix.
In general, a square complex matrix A is similar to a block diagonal matrix = [] where each block J i is a square matrix of the form = []. So there exists an invertible matrix P such that P −1 AP = J is such that the only non-zero entries of J are on the diagonal and the superdiagonal.
The spectral theorem for normal matrices is a special case of the more general Schur decomposition which holds for all square matrices. Let A be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B. If A is normal, so is B.
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: a + i b ≡ ...
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...