Math 2.0

Linear and multilinear algebra; matrix theory Definition

Similar matrices

Builds on Invertible matrix

Definition

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

Remarks

Similarity is an equivalence relation on Mn(F)M_n(F). By the change of basis formula, two matrices are similar if and only if they represent the same endomorphism of FnF^n in two bases. Similar matrices have the same trace, determinant, characteristic polynomial, rank and eigenvalues. The matrices I2I_2 and (1101)\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} have the same characteristic polynomial but are not 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 matrixdefinitionSimilar matrices, definition, no evidence levelSimilar matricesdefinition
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  1. Identity matrix object
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Further reading

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