Gram-Schmidt
Vector independence procedure
Gram-Schmidt is a mathematical procedure for converting a set of vectors into an orthogonal or orthonormal set—vectors that are mutually perpendicular and, if normalized, of unit length.
The process works iteratively: each vector is adjusted by subtracting its projections onto the previously processed vectors.
Apply Gram-Schmidt to find an orthonormal basis.
Named for mathematicians Jørgen Gram and Erhard Schmidt.
Matrix as orthogonal-triangular product
- basisUnderlying support or principle
- vectorsDirected magnitude quantities
- linear algebraMathematics of vector spaces
- projectionSomething that extends outward
normalization · span · coefficients · iterative · decomposition · factorization · least squares · matrix
Error-minimizing statistical method
- signal processingMathematical manipulation of waveforms
- image compressionShrinking picture file sizes
- data scienceExtracting meaning from information
- machine learningAlgorithms that improve from data
regression · curve fitting · noise reduction · wireless communications · computer graphics · crystallography · quantum mechanics
Changed in some way
- classicalOf ancient Greece/Rome
- stabilityResistance to displacement
- rounding errorInaccuracy from approximating numbers
- floating-pointDecimal number representation
precision · accumulation · cancellation · ill-conditioned · pivoting
QR decomposition
Matrix as orthogonal-triangular product
- basisUnderlying support or principle
- vectorsDirected magnitude quantities
- linear algebraMathematics of vector spaces
projection · normalization · span · coefficients · iterative · decomposition · factorization · least squares · matrix · subspace
least squares
Error-minimizing statistical method
- signal processingMathematical manipulation of waveforms
- image compressionShrinking picture file sizes
- data scienceExtracting meaning from information
machine learning · regression · curve fitting · noise reduction · wireless communications · computer graphics · crystallography · quantum mechanics
modified
Changed in some way
- classicalOf ancient Greece/Rome
- stabilityResistance to displacement
- rounding errorInaccuracy from approximating numbers
floating-point · precision · accumulation · cancellation · ill-conditioned · pivoting
- functional analysisnorm, norms, completeness, orthogonality, subspaces
- linear algebramatrix, linear equation, vectors, linear function, matrices
- QR factorizationHouseholder
- mathematical functionssine function, exponential function, logarithm, trigonometric functions, linear functions
- DecompositionsLU decomposition, QR decomposition, eigendecomposition, diagonalization, prime factorization
gram, Schmidt, grams, grammar, grammarian, grammatical, grammatically, program, diagram