Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Gram matrices are positive semidefinite

Builds on Positive semidefinite matrixConjugate transpose

Statement

Let BB be an m×nm \times n complex matrix. Then the n×nn \times n matrix B∗BB^* B is positive semidefinite, where B∗B^* is the conjugate transpose. It is positive definite if and only if the columns of BB are linearly independent.

Remarks

The proof: x∗B∗Bx=(Bx)∗(Bx)=∥Bx∥2≥0x^* B^* B x = (Bx)^* (Bx) = \norm{Bx}^2 \ge 0. For a real matrix BB, the same holds for BTBB^T B. Each Gram matrix has this form, so Gram matrices are positive semidefinite.

Reading path

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Matrix, definition, no evidence levelMatrixdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionPositive semidefinite matrix, definition, no evidence levelPositive semidefinite…definitionGram matrices are positive semidefinite, statement, no evidence levelGram matrices are pos…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Matrix definition
  2. Square matrix definition
  3. Transpose definition
  4. Conjugate transpose definition
  5. Hermitian matrix definition
  6. Positive semidefinite matrix definition
  7. Gram matrices are positive semidefinite statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Definite_matrix, Decomposition
About this page
Tag0AM050KCAW
URIm2://0AM050KCAW
Kindstatement, proposition
WritingW0 stub
MSC15B57, 15A63
StatusThis copy of the site has no build data. Run m2 status 0AM050KCAW.
UsesPositive semidefinite matrix, Conjugate transpose

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