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Matrix equivalence

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Definition of equivalent matrices
A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.
Two matrices of the same size are equivalent if and only if they have the same rank.
Equivalent matrices represent the same linear transformation with resect to appropriate bases.

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Equivalent matrices represent the same linear transformation with resect to appropriate bases.

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Definition of equivalent matricesClose match
Matrix equivalenceClose match
Similarity of matricesClose match
Equivalence theorem for nonsingular matrices: the matrix A is a change-of-basis matrix.Close match
Two matrices of the same size are equivalent if and only if they have the same rank.Close match
Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.Close match
Definition of similar matricesClose match
Definition of row equivalent matricesClose match
Similarity of matrices is an equivalence relationClose match
Equivalence theorem for injective linear transformations: The columns of the matrix of T are linearly independent.Close match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is an isomorphism.Close match
Row-equivalent augmented matrices represent equivalent systemsClose match
A matrix equation is equivalent to a linear systemClose match
Equivalence theorem for nonsingular matrices: the columns of A are a basis for R^n (or C^n).Close match
Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).Close match
Definition of equality of matricesClose match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax has an inverse.Close match
Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.Close match
Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.Close match
Equivalence theorem for nonsingular matrices: the matrix A has an inverse.Close match

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