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16 · Matrices

m×n matrix

m×n matrix ALG-MAT-001 intermediate Linengular array of scalars. ALG-MAT-002 ALG-SIS-002 matrix row–column structure of a matrix. A=(a ij ) m n

16 · Matrices

Matrix addition

Matrix addition ALG-MAT-002 intermediate Entrywise sum for matrices of the same size. ALG-MAT-003 matrix entrywise sum. (A+B) ij =a ij +b ij

16 · Matrices

Scalar multiplication

Scalar multiplication ALG-MAT-003 intermediate Scales every entry of the matrix. ALG-MAT-004 matrix uniform scaling of all entries. (cA) ij =ca ij

16 · Matrices

Matrix product

Matrix product ALG-MAT-004 intermediate Each entry arises from a row–column dot product. ALG-TRA-002 ALG-DEC-001 matrix row-by-column rule for the matrix product. (AB) ij = k a ik b kj El número de columnas de

16 · Matrices

Matrix product El número de columnas de A debe coincidir with el número de filas de B. En general, AB\ne BA.

16 · Matrices

Identity matrix

Identity matrix ALG-MAT-005 intermediate Identity element for matrix multiplication. ALG-DET-005 matrix identity as the transformation that does not change a vector or matrix. AI=IA=A

16 · Matrices

Transpose

Transpose ALG-MAT-006 intermediate Swaps rows and columns. ALG-ORT-003 ALG-LSQ-002 matrix transposition as swapping rows and columns. (A T) ij =a ji ; (AB) T=B TA T

16 · Matrices

Symmetric matrix

Symmetric matrix ALG-MAT-007 intermediate Square matrix equal to its transpose. ALG-EIG-006 ALG-DEC-003 matrix matrix symmetry with respect to the main diagonal. A T=A

17 · Determinants and inverses

Invertibility criterion

Invertibility criterion ALG-DET-004 intermediate Characterizes invertible square matrices. ALG-DET-005 ALG-ESP-002 geometry invertibility as preservation of nonzero dimension/area. A invertible (A) 0

19 · Linear transformations

Linear composition

Linear composition ALG-TRA-005 intermediate Composing transformations corresponds to multiplying matrices in the proper order. ALG-FUN-002 ALG-MAT-004 matrix_transform composición como aplicación sucesiva de dos transformacio

20 · Eigenvalues and diagonalization

Powers via diagonalization

Powers via diagonalization ALG-EIG-005 advanced Simplifies matrix powers and recurrences. ALG-EIG-004 ALG-SEC-007 matrix_transform powers of a matrix via scalar powers of eigenvalues. A n=PD nP

23 · Matrix decompositions

LU decomposition

LU decomposition ALG-DEC-001 advanced Factors a matrix into triangular factors, with pivoting when needed. ALG-SIS-004 matrix LU as a compact sequence of Gaussian elimination. PA=LU

23 · Matrix decompositions

QR decomposition

QR decomposition ALG-DEC-002 advanced Q has orthonormal columns and R is upper triangular. ALG-ORT-003 ALG-ORT-004 ALG-LSQ-001 matrix_transform QR as an orthonormal basis plus triangular coordinates. A=QR

23 · Matrix decompositions

Spectral decomposition

Spectral decomposition ALG-DEC-003 advanced Spectral form of real symmetric matrices. ALG-EIG-006 matrix_transform spectral decomposition of a symmetric matrix. A=Q Q T

23 · Matrix decompositions

SVD

SVD ALG-DEC-004 advanced Universal decomposition into singular directions and scales. ALG-LSQ-003 ALG-DEC-005 ALG-NOR-002 ALG-NOR-005 ALG-NOR-007 matrix_transform SVD como rotación/reflexión → escalamiento → rotación/re

23 · Matrix decompositions

SVD - compresión de datos e imágenes; - reducción dimensional; - pseudoinversa y mínimos cuadrados; - sistemas de recomendación; - análisis de datos y aprendizaje automático.

23 · Matrix decompositions

Low-rank approximation

Low-rank approximation ALG-DEC-005 advanced Approximates a matrix using its k dominant singular values. ALG-DEC-004 ALG-ESP-005 matrix_transform aproximación de bajo rango eliminando componentes singulares pequeñ

29 · Relationship maps

Vectors → Linear combination → Span → Independencia → Base → Dimensión Matrices → Sistemas → Gauss → Rank/Nullity Transformaciones → Kernel/Image → Valores propios → Diagonalization Dot product → Orthogonality → Pr

30 · Boundaries with other subjects

Precálculo / Trigonometría: identidades trigonométricas, ecuaciones trigonométricas, coordenadas polares y cónicas extensas. Cálculo: límites, derivadas, integrales, series generales y ecuaciones diferenciales. Matemátic