Math 2.0

Linear and multilinear algebra; matrix theory Definition

Equivalent matrices

Builds on Invertible matrix

Definition

Let FF be a field and let AA and BB be m×nm \times n matrices over FF. The matrix BB is equivalent to AA when there are an invertible n×nn \times n matrix PP and an invertible m×mm \times m matrix QQ over FF with B=Q−1AP.B = Q^{-1} A P.

Remarks

Two matrices are equivalent if and only if they represent the same linear map Fn→FmF^n \to F^m in two pairs of bases. Two m×nm \times n matrices over a field are equivalent if and only if they have the same rank. A matrix of rank rr is equivalent to the block matrix (Ir000)\begin{pmatrix} I_r & 0 \\ 0 & 0 \end{pmatrix}. Similarity is a finer relation: similar matrices are equivalent, but equivalent square matrices need not be similar.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionInvertible matrix, definition, no evidence levelInvertible matrixdefinitionEquivalent matrices, definition, no evidence levelEquivalent matricesdefinition
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  1. Identity matrix object
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  5. Invertible matrix definition
  6. Equivalent matrices definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Matrix_equivalence, lead section
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