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The Rank of a Matrix
Contents:
  1. Linear Algebra C-3 | ZODML
  2. Search form
  3. Presentation:
  4. R Companion to Linear Algebra Step by Step, part 2
  5. Surface Area

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Linear Algebra C-3 | ZODML

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Search form

DCOPY - copy x into y. DSDOT - dot product with extended precision accumulation. DNRM2 - Euclidean norm.


  • Linear Algebra C-3.
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DASUM - sum of absolute values. CSWAP - swap x and y. CCOPY - copy x into y. CDOTU - dot product.


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CDOTC - dot product, conjugating the first vector. ZSWAP - swap x and y. ZCOPY - copy x into y.

Linear Algebra - [The Gram-Schmidt Process] - OrthoNormalizing Vectors (C^3 Example 2)

ZDOTU - dot product. ZDOTC - dot product, conjugating the first vector. SGEMV - matrix vector multiply. SGBMV - banded matrix vector multiply. SSYMV - symmetric matrix vector multiply. SSBMV - symmetric banded matrix vector multiply. SSPMV - symmetric packed matrix vector multiply.

Presentation:

STRMV - triangular matrix vector multiply. STBMV - triangular banded matrix vector multiply. STPMV - triangular packed matrix vector multiply. STRSV - solving triangular matrix problems.

STBSV - solving triangular banded matrix problems. STPSV - solving triangular packed matrix problems. DGEMV - matrix vector multiply. DGBMV - banded matrix vector multiply. DSYMV - symmetric matrix vector multiply. DSBMV - symmetric banded matrix vector multiply. DSPMV - symmetric packed matrix vector multiply. DTRMV - triangular matrix vector multiply. DTBMV - triangular banded matrix vector multiply.

DTPMV - triangular packed matrix vector multiply. DTRSV - solving triangular matrix problems. DTBSV - solving triangular banded matrix problems. DTPSV - solving triangular packed matrix problems. CGEMV - matrix vector multiply. CGBMV - banded matrix vector multiply. CHEMV - hermitian matrix vector multiply. CHBMV - hermitian banded matrix vector multiply. CHPMV - hermitian packed matrix vector multiply. CTRMV - triangular matrix vector multiply. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.

Corporate eLibrary

Consider matrix A and its row echelon matrix, A ref. Previously, we showed how to find the row echelon form for matrix A.

R Companion to Linear Algebra Step by Step, part 2

Because the row echelon form A ref has two non-zero rows, we know that matrix A has two independent row vectors; and we know that the rank of matrix A is 2. You can verify that this is correct. Row 1 and Row 2 of matrix A are linearly independent. However, Row 3 is a linear combination of Rows 1 and 2. Therefore, matrix A has only two independent row vectors.

Surface Area

When all of the vectors in a matrix are linearly independent , the matrix is said to be full rank. Consider the matrices A and B below. Notice that row 2 of matrix A is a scalar multiple of row 1; that is, row 2 is equal to twice row 1. Therefore, rows 1 and 2 are linearly dependent.