Diagonalizable Matrix

A Diagonalizable Matrix is a square matrix, A, such that there exists an invertible matrix P where P⁻¹AP is a diagonal matrix.

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• Example(s):
  • a Diagonalizable Covariance Matrix (covariance matrix).
• See: Homogeneous Dilation, Similar (Linear Algebra), Dimension (Linear Algebra), Vector Space, Linear Map, Defective Matrix, Transformation Matrix.

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References

2016

• (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/diagonalizable_matrix Retrieved:2016-4-13.
  • In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P⁻¹AP is a diagonal matrix. If V is a finite-dimensional vector space, then a linear map T : V → V is called diagonalizable if there exists an ordered basis of V with respect to which T is represented by a diagonal matrix. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. A square matrix that is not diagonalizable is called defective.

    Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle: their eigenvalues and eigenvectors are known and one can raise a diagonal matrix to a power by simply raising the diagonal entries to that same power. Geometrically, a diagonalizable matrix is an inhomogeneous dilation (or anisotropic scaling)  — it scales the space, as does a homogeneous dilation, but by a different factor in each direction, determined by the scale factors on each axis (diagonal entries).

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