Math 2.0

Linear and multilinear algebra; matrix theory Definition

Scalar matrix

Builds on Diagonal matrixIdentity matrix

Definition

Let RR be a ring, let n≥0n \ge 0 be an integer, and let c∈Rc \in R. The scalar matrix of order nn with value cc is the diagonal matrix diag⁡(c,…,c)=cIn\operatorname{diag}(c, \dots, c) = c I_n, where InI_n is the identity matrix. A square matrix is a scalar matrix when it equals cInc I_n for some c∈Rc \in R.

Remarks

Over a commutative ring, a scalar matrix commutes with every square matrix of the same order. Over a field, the scalar matrices are exactly the n×nn \times n matrices that commute with every n×nn \times n matrix. As a linear map, cInc I_n multiplies each vector by cc.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionDiagonal matrix, definition, no evidence levelDiagonal matrixdefinitionScalar matrix, definition, no evidence levelScalar matrixdefinition
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  1. Identity matrix object
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  3. Square matrix definition
  4. Diagonal matrix definition
  5. Scalar matrix definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Diagonal_matrix, Scalar matrix
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DefinesScalar matrix
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