linear algebra, a square matrix A {\displaystyle A}  is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists...

factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can...

contradict what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation...

rotation matrices. A method for finding log A for a diagonalizable matrix A is the following: Find the matrix V of eigenvectors of A (each column of V is an...

there exists a unitary matrix U such that UAU* and UBU* are diagonal matrices. In other words A and B are simultaneously diagonalizable. In this special case...

where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called diagonalizable. More generally, and...

{\displaystyle AB=BA} ) if they are simultaneously diagonalizable (that is, there exists an invertible matrix P {\displaystyle P} such that both P − 1 A P {\displaystyle...

algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an...

∗ {\displaystyle U^{*}U=UU^{*}} ). U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem...

matrix B. For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex...

known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A {\displaystyle A} , the algorithm will produce a number λ {\displaystyle...

nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible. The only nilpotent diagonalizable matrix is the zero matrix. See also:...

decomposition) of a diagonalizable matrix is a decomposition of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in...

P is the minimal polynomial of A. We further assume that A is a diagonalizable matrix. In particular, the roots of P are simple, and the "interpolation"...

The matrix function f ( X ) {\displaystyle f(\mathbf {X} )} is defined in terms of the scalar function f ( x ) {\displaystyle f(x)} for diagonalizable matrices...

\in \mathbb {C} ^{n}.} A square matrix A {\displaystyle A} is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues. Hermitian...

vector space of dimension d {\displaystyle d} over a field is a diagonalizable matrix, since its minimal polynomial divides x 2 − x {\displaystyle x^{2}-x}...

diagonal entries of a matrix. Anti-diagonal matrix Banded matrix Bidiagonal matrix Diagonally dominant matrix Diagonalizable matrix Jordan normal form Multiplication...

diagonalization procedure. A diagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all 1 × 1. More...

general (complex) symmetric matrix may be defective and thus not be diagonalizable. If A {\displaystyle A} is diagonalizable it may be decomposed as A =...

computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization...

In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices...

quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity. In the theory of algebraic...

a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a...

C(p)} is diagonalizable as C ( p ) = V − 1 D V {\displaystyle C(p)=V^{-1}\!DV} , where D is the diagonal matrix and V is the Vandermonde matrix corresponding...

Gram matrix over the reals is a symmetric matrix, it is diagonalizable and its eigenvalues are non-negative. The diagonalization of the Gram matrix is the...

skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their...

Scaling in the most general sense is any affine transformation with a diagonalizable matrix. It includes the case that the three directions of scaling are not...

matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...

triangular matrix are precisely its eigenvalues (see Triangular matrix#Properties). An n × n matrix A is diagonalizable if there is a matrix V and a diagonal...