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Similarity, Similar matrices, Diagonable matrices, Orthogonal similarity, Real quadratic forms, Hermitian matrices, Normal matrices

Similar matrices. Two n-square matrices A and B over a field F are called similar if there exists a non-singular matrix P over F such that

(1) B = P^{ -1}AP

Note. Note that if B = P^{-1}AP then A = PBP^{-1} . The condition of similarity can be defined by
either formula. The formula B = P^{-1}AP becomes B = QAQ^{ -1} if we let Q = P^{-1}.

Similarity is an equivalence relation that separates the set of all n-square matrices into equivalence classes. All matrices similar to a given matrix are similar to each other. What is the significance of this relationship of similarity between matrices? The answer is that any matrix similar to a given matrix represents the same linear transformation as the given matrix, but as referred to a different coordinate system (or basis). Thus any two matrices that are similar to each other represent the same point transformation in n-space i.e. they map points in the same way, they represent the same linear point transformation.

The concept of similarity is thus intricately connected to the concept of a change in basis, a change in coordinate system. Changing the basis for a linear transformation produces similar matrices.

Changes in the expression of a linear transformation due to a change in basis. Let Y = AX be a linear point transformation expressed with respect to the E-basis. What is the expression for this same transformation when expressed with respect to some arbitrary other Z-basis? In the Z-basis the vectors X and Y are

X = ZX_{Z} Y = ZY_{Z} .

Substituting into Y = AX we get

ZY_{Z} = AZX_{Z}

or

Y_{Z} = Z^{-1}AZX_{Z} .

Thus we see that the matrix Z^{-1}AZ of the point transformation with respect to the Z-basis is
similar to the matrix A of the transformation with respect to the E-basis.

All matrices that are similar to each other represent the same linear point transformation, but as referred to different frames of reference, or basis. Of great importance are those n-square matrices that are similar to diagonal matrices. Any matrix that is similar to a diagonal matrix represents the same linear point transformation as the diagonal matrix. What are the characteristics of the linear point transformation effected by a diagonal matrix? The answer is as follows.

Point transformation effected by a diagonal matrix. The point transformations effected by a diagonal matrix represent a very important class of linear transformations. Given the diagonal matrix

consider the point transformation effected by it in n-space (i.e. the transformations given by Y =
DX ). The transformations of this class have a very simple and intuitive geometric meaning (of
course, only for two and three-dimensional real spaces). If the numbers k_{i} are all positive the
effect of this transformation is simply stretching (or compressing) effects directed in the
directions of the different coordinate system axes e_{1}, e_{2}, .... ,e_{n} with magnitudes given by k_{1}, k_{2},
.... ,k_{n}. If some of the k_{i} are negative, then the deformation of the space is accompanied by a
change of direction. Finally, if for example k_{1 }= 0, then a projection of the space parallel to e_{1}
takes place onto the subspace spanned by e_{2}, e_{3}, .... ,e_{n} with a subsequent deformation in these
directions. If the coordinate system is an oblique one the transformation will transform figures in
space in this linear way in the directions of the oblique axes.

This class of transformations is important because in spite of its simplicity it is very general. In fact, it can be established that every linear transformation satisfying certain not very severe restrictions belongs to this class i.e. we can find for it a basis in which it is described by a diagonal matrix. [ excerpted from Mathematics, Its Content, Methods and Meaning. Vol. III. p. 80]

Diagonable matrices. A mattrix A which is similar to a diagonal matrix is called diagonable.

Under what conditions is an n-square matrix similar to a diagonal matrix? The following theorems address that question.

Theorems.

1] If all the eigenvalues of a matrix are distinct, then the matrix is similar to a diagonal matrix whose diagonal elements are these eigenvalues.

2] If an n-square matrix has n linearly independent eigenvectors, it is similar to a diagonal matrix.

3] An n-square matrix A is similar to a diagonal matrix if and only if the eigenvectors of A span
V_{n}(F); when this is the case, the eigenvalues of A are the diagonal elements of D.

4] An n-square matrix A is similar to a diagonal matrix if and only if for each eigenvalue λ_{i} of A,
the multiplicity of λ_{i} is equal to the dimension of the null space of λ_{i}I - A.

5] Two diagonal matrices are similar if and only if they differ only in the order of their diagonal elements.

6] Let A be an n-square matrix with n linearly independent eigenvectors p_{1}, p_{2}, ... , p_{n} and let λ_{1},
λ_{2}, ... , λ_{n} be the eigenvalues corresponding to p_{1}, p_{2}, ... , p_{n}. Let P be a matrix whose columns 1,
2, ... , n consist of p_{1}, p_{2}, ... , p_{n} i.e.

P = [p_{1} p_{2} ... p_{n}]

and let D be the diagonal matrix

whose diagonal elements consist of the eigenvalues λ_{1}, λ_{2}, ... , λ_{n}. Then matrix P is non-singular
and

2) A = PDP^{-1}

or, equivalently,

3) D = P^{-1}AP .

Significance of Theorem 6. What is the significance of Theorem 6? Consider the linear point transformation

Y = AX

where matrix A has n distinct eigenvalues and thus also n corresponding linearly independent
eigenvectors. We would like to know something about the nature of this linear point
transformation effected by matrix A, just how it transforms figures in n-space. Could we perhaps
get some insight into how it transforms figures by doing a change of coordinate system,
transferring over to the coordinate system defined by the eigenvectors? Let us do a change of
coordinate system, a change of basis, transferring over to the eigenvector coordinate system.
From Theorem 6 we know that if Y = AX then Y = PDP^{-1}X where P is a matrix whose columns
consist of the eigenvectors and D is a diagonal matrix whose diagonal elements consist of the
corresponding eigenvalues. If X' and Y' are the values of X and Y in the eigenvector coordinate
system the equations relating the coordinates in the two systems are

X = PX'

Y = PY' .

Substituting into Y = PDP^{-1}X we get

PY' = PDP^{-1}PX'

or, equivalently,

4) Y' = DX' .

Thus in the eigenvector coordinate system the linear point transformation Y = AX is given by Y' = DX' , a transformation effected by the diagonal matrix D. What does this tell us? It tells us that the transformation of the figure by Y = AX consists of a distortion consisting of stretching (or compressing) in the directions of the various eigenvector axes by amounts given by the associated eigenvalues. It gives us an intuitive insight into the linear point transformation Y = AX.

Similar matrices and eigenvalues.

7] Two similar matrices have the same eigenvalues.

Relationship between the eigenvectors of two similar matrices. Let X_{i}
be an eigenvector of a matrix A corresponding to an eigenvalue λ_{i} of A. Then an eigenvector Y_{i}
of a similar matrix B = P ^{-1}AP corresponding to the same eigenvalue λ_{i} of B is given by Y_{i} = P^{ -1}X.

Eigenvalues and eigenvectors of diagonal matrices.

8] The eigenvalues of a diagonal matrix D = diag(a_{1}, a_{2}, .... ,a_{n}) are simply the diagonal
elements.

9] A diagonal matrix D = diag(a_{1}, a_{2}, .... ,a_{n}) always has n linearly independent eigenvectors.
The elementary unit vectors

..................

constitute a set of n linearly independent eigenvectors for D since DE_{i} = a_{i}E_{i}, ( i =1, 2, ...., n).

General theorems.

10] Any n-square matrix A, similar to a diagonal matrix, has n linearly independent eigenvectors.

11] Every square matrix A is similar to a triangular matrix whose diagonal elements are the eigenvalues of A.

12] If A is any real n-square matrix with real eigenvalues, there exists an orthogonal matrix Q
such that Q ^{-1}AQ = Q^{T}AQ is triangular and has as diagonal elements the eigenvalues of A The
matrices A and Q ^{-1}AQ are called orthogonally similar.

13] If A is any n-square matrix with complex elements or a real n-square matrix with complex eigenvalues, there exists a unitary matrix U such that is triangular and has as diagonal elements the eigenvalues of A. The matrices A and are called unitarily similar.

Synonyms.

Eigenvalue, characteristic root, latent root

Eigenvector, characteristic vector, invariant vector, latent vector

Eigenvector space, characteristic vector space, invariant vector space

Real symmetric matrices.

1] The characteristic roots (i.e. eigenvalues) of a real symmetric matrix are all real.

2] The invariant vectors (i.e. eigenvectors) associated with distinct characteristic roots of a real symmetric matrix are mutually orthogonal.

3] If A is a real n-square symmetric matrix with characteristic roots λ_{1}, λ_{2}, ... , λ_{n}, then there
exists a real orthogonal matrix P such that P^{T}AP = P^{ -1}AP = diag(λ_{1}, λ_{2}, ... , λ_{n}).

4] If λ_{i} is a characteristic root of multiplicity r_{i} of a real symmetric matrix, then there is
associated with λ_{i} an invariant space of dimension r_{i} .

Orthogonal similarity. If P is an orthogonal matrix and B = P^{ -1}AP. then B is said to be
orthogonally similar to A. Since P^{ -1} = P^{T}, B is also orthogonally congruent and orthogonally
equivalent to A.

Every real symmetric matrix A is orthogonally similar to a diagonal matrix whose diagonal elements are the characteristic roots of A.

Canonical set for real symmetric matrices under orthogonal
similarity. Let the characteristic roots of the real symmetric matrix A be arranged so that λ_{1}
≥ λ_{2} ≥.... ≥ λ_{n} . Then diag( λ_{1}, λ_{2}, ... , λ_{n}) is a unique diagonal matrix similar to A. The totality of
such diagonal matrices constitutes a canonical set for real symmetric matrices under orthogonal
similarity.

Two real symmetric matrices are orthogonally similar if and only if they have the same characteristic roots, that is if and only if they are similar.

Real quadratic forms.

1] Every real quadratic form q = X^{T}AX can be reduced by an orthogonal transformation X =
BY to a canonical form

where r is the rank of A and λ_{1}, λ_{2}, ... , λ_{n} are its non-zero characteristic roots.

2] A real symmetric matrix is positive definite if and only if all of its characteristic roots are positive.

3] If X^{T}AX and X^{T}BX are real quadratic forms in (x_{1}, x_{2}, .... ,x_{n}) and if X^{T}BX is positive
definite, there exists a real non-singular linear transformation X = CY which carries X^{T}AX into

and X^{T}BX into

where λ_{i} are the roots of |λB - A| = 0 .

Hermitian matrices.

1] The characteristic roots of an Hermitian matrix are real.

2] The invariant vectors associated with distinct characteristic roots of an Hermitian matrix are mutually orthogonal.

3] If H is an n-square Hermitian matrix with characteristic roots λ_{1}, λ_{2}, ... , λ_{n}, there exists a
unitary matrix
such that
= diag(λ_{1}, λ_{2}, ... , λ_{n} ) . The matrix H is called
unitarily similar to
.

4] If λ_{i} is a characteristic root of multiplicity r_{i} of the Hermitian matrix H, then there is
associated with λ_{i} an invariant space of dimension r_{i} .

Canonical set for Hermitian matrices under unitary similarity. Let the
characteristic roots of the Hermitian matrix H be arranged so that λ_{1} ≤ λ_{2} ≤ .... λ_{n}. Then diag(λ_{1},
λ_{2}, ... , λ_{n}) is a unique diagonal matrix similar to H. The totality of such diagonal matrices
constitutes a canonical set for Hermitian matrices under unitary similarity.

Two Hermitian matrices are unitarily similar if and only if they have the same characteristic roots, that is if and only if they are similar.

Normal matrices. An n-square matrix A is called normal if . Normal matrices include diagonal, real symmetric, real skew-symmetric, orthogonal, Hermitian, skew-Hermitian and unitary matrices.

1] If is a normal matrix and is a unitary matrix, then is a normal matrix.

2] If X_{i} is an invariant vector corresponding to the characteristic root λ_{i} of a normal matrix A,
then X_{i} is also an invariant vector of
corresponding to the characteristic root
.

3] A square matrix A is unitarily similar to a diagonal matrix if and only if A is normal.

4] If A is normal, the invariant vectors corresponding to distinct characteristic roots are orthogonal.

5] If λ_{1} is a characteristic root of multiplicity r_{i} of a normal matrix A, the associated invariant
space has dimension r_{i} .

6] Two normal matrices are unitarily similar if and only if they have the same characteristic roots, that is if and only if they are similar.

References.

1. Ayres. Matrices (Schaum).

2. Mathematics, Its Content, Methods and Meaning. Vol. III

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