Math 2.0

definition

Singular value

Definition

Let AA be an m×nm \times n complex matrix. The matrix A∗AA^* A is positive semidefinite, so its eigenvalues λ1≥⋯≥λn≥0\lambda_1 \ge \cdots \ge \lambda_n \ge 0 are real and nonnegative. The singular values of AA are the numbers σi=λi(1≤i≤n).\sigma_i = \sqrt{\lambda_i} \qquad (1 \le i \le n).

Remarks

The number of nonzero singular values is the rank of AA. The largest singular value is the operator norm max⁡∥x∥=1∥Ax∥\max_{\norm{x} = 1} \norm{Ax}. For a normal matrix, the singular values are the absolute values of the eigenvalues. Some authors list only the first min⁡(m,n)\min(m, n) values.

Explanations

L1 human

The singular values of a matrix measure how much it stretches space in its main directions. The unit sphere goes to an ellipsoid, and the singular values are the lengths of its semi-axes. Unlike eigenvalues, they exist for every matrix, also rectangular ones.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionPositive semidefinite matrix, definition, no evidence levelPositive semidefinite…definitionGram matrices are positive semidefinite, statement, no evidence levelGram matrices are pos…statementSingular value, definition, no evidence levelSingular valuedefinition
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Matrix definition
  2. Vector space definition
  3. Square matrix definition
  4. Transpose definition
  5. Linear map definition
  6. Conjugate transpose definition
  7. Endomorphism definition
  8. Hermitian matrix definition
  9. Eigenvalue definition
  10. Positive semidefinite matrix definition
  11. Gram matrices are positive semidefinite statement
  12. Singular value definition

What depends on this

1 node uses this node, directly or through other nodes.

Sources

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

What links here

Used by