Degenerate matrix

Degenerate matrix (synonyms: singular matrix , special matrix , special matrix ) - square matrix $A,$ ${\ displaystyle A,}$ $A$ determinant of which $\det(A)$ ${\ displaystyle \ det (A)}$ $\ det (A)$ equal to zero.

Content

Equivalent degeneracy conditions

Using various concepts of linear algebra , one can give various degeneracy conditions:

Rows or columns of a matrix are linearly dependent . In other words, in a degenerate matrix there are at least two rows (or two columns) ${\bf {{x}_{i}}}$ ${\ displaystyle {\ bf {{x} _ {i}}}}$ and ${\bf {{x}_{j},}}$ ${\ displaystyle {\ bf {{x} _ {j},}}}$ meeting the condition $a{\bf {{x}_{i}={\bf {{x}_{j},}}}}$ ${\ displaystyle a {\ bf {{x} _ {i} = {\ bf {{x} _ {j},}}}}}$ where a is a scalar . In particular, any square matrix containing a zero column or row is degenerate.
Square matrix $A$ ${\ displaystyle A}$ degenerate if and only if there is a nonzero vector $x,$ ${\ displaystyle x,}$ such that $Ax=0.$ ${\ displaystyle Ax = 0.}$ In other words, the linear operator corresponding to the matrix in the standard basis has a nonzero kernel .
Square matrix $A$ ${\ displaystyle A}$ degenerate if and only if it has at least one zero eigenvalue $\lambda =0.$ ${\ displaystyle \ lambda = 0.}$ This follows from the equation, which is satisfied by all the eigenvalues of the matrix: $\det(A-\lambda E)=0$ ${\ displaystyle \ det (A- \ lambda E) = 0}$ (where E is the identity matrix ), as well as from the fact that the determinant of a matrix is equal to the product of its eigenvalues.

Properties

A degenerate matrix does not have a standard inverse matrix . At the same time, a degenerate matrix has a pseudoinverse matrix (generalized inverse matrix) or even an infinite number of them.
The rank of a degenerate matrix is less than its size (number of rows).
The product of a degenerate matrix and any square matrix with the same size gives a degenerate matrix. This follows from the property $\det(AB)=\det(A)\cdot \det(B).$ ${\ displaystyle \ det (AB) = \ det (A) \ cdot \ det (B).}$ A degenerate matrix raised to any integer positive degree remains degenerate. A product of any number of matrices is degenerate if and only if at least one of the factors is degenerate. The product of non-degenerate matrices cannot be degenerate.
The transposition of a degenerate matrix leaves it degenerate (since transposition does not change the determinant of the matrix, $\det(A^{T})=\det(A)$ ${\ displaystyle \ det (A ^ {T}) = \ det (A)}$ )
Multiplying a degenerate matrix by a scalar leaves it degenerate (since $\det(\alpha A)=\alpha ^{n}\det(A)=0$ ${\ displaystyle \ det (\ alpha A) = \ alpha ^ {n} \ det (A) = 0}$ , where n is the size of the degenerate matrix A , α is the scalar).
The Hermitian-conjugate matrix of a degenerate matrix is degenerate (since the determinant of a Hermitian-conjugate matrix is complex conjugate with the determinant of the original matrix and, therefore, is equal to zero).
The union (mutual, adjoint) matrix of a degenerate matrix is degenerate (this follows from the property of union matrices $\det(\operatorname {adj} (A))=(\det A)^{n-1}$ ${\ displaystyle \ det (\ operatorname {adj} (A)) = (\ det A) ^ {n-1}}$ ) The product of a degenerate matrix by its union matrix gives a zero matrix : $A\cdot \operatorname {adj} (A)=\operatorname {adj} (A)\cdot A=0,$ ${\ displaystyle A \ cdot \ operatorname {adj} (A) = \ operatorname {adj} (A) \ cdot A = 0,}$ since for an arbitrary square matrix $A\cdot \operatorname {adj} (A)=\operatorname {adj} (A)\cdot A=\det(A)\cdot E.$ ${\ displaystyle A \ cdot \ operatorname {adj} (A) = \ operatorname {adj} (A) \ cdot A = \ det (A) \ cdot E.}$
A triangular (and, in particular, diagonal ) matrix is degenerate if and only if at least one of its elements on the main diagonal is zero. This follows from the fact that the determinant of a triangular matrix is equal to the product of elements on its main diagonal.
If the matrix A is degenerate, then the system of equations $Ax=0$ ${\ displaystyle Ax = 0}$ has nonzero solutions.
Rearrangement of rows or columns of a degenerate matrix gives a degenerate matrix.
A degenerate matrix, considered as a linear operator , maps a vector space into its subspace of lower dimension.

Special cases

Degenerate matrices are, in particular:

zero matrix (consisting of only zeros);
matrix of units (consisting of one unit) with size n > 1 ;
nilpotent matrices (matrices whose some natural degree is a zero matrix);
- shift matrices (a subset of nilpotent matrices);
Vandermonde matrix , if at least two of its parameters coincide;
Gell-Mann matrices in standard representation (except for λ ₈ );
The Kirchhoff matrix (also known as the Laplace matrix) is the matrix representation of the graph.

Literature

Gantmakher F.R. Matrix Theory (2nd ed.). M .: Science , 1966
Lancaster P. Matrix Theory M .: Science , 1973