Rank of a Matrix

3 Key excerpts on "Rank of a Matrix"

• 

  eBook - PDF

  Linear Algebra

  Examples and Applications

  • Alain M Robert(Author)
  • 2005(Publication Date)
  • WSPC

    (Publisher)

  Chapter 5 The Rank Theorem The abstract language of linear algebra has allowed us to prove the invariance of the rank, as well as the equalities between row- and column-rank. This is the rank theory, which has far-reaching consequences: Some of them are presented in this chapter. On the other hand, we give further examples showing how the general concept of vector does enrich the mathematical description of natural phenomena: p A velocity (vector) is more precise than a speed (scalar) p A colored pixel is richer than a grey one p A pyramid of ages contains more information than a bare total population figure. 5.1 More on Row- versus Column-Rank We have already seen that the row-rank r = dimL(rows of A) is the same as the column-rank dim L(co1umns of A) = dim im(A). In other words, the ranks of a matrix A and its transpose % are the same. More can be said. 5.1.1 Factorizations of a Matrix Proposition. Any matrix A of size m x n and rank r admits a factorization A = ST where the size of S is m x r and the size of T is r x n. PROOF. Proceeding with row operations, we can find an echelon form of the matrix A, say A - U = EA, where E is invertible of size m x m * * ... ... ~ [mi ... ... 116 5.1. MORE ON ROW- VERSUS COLUMN-RANK 117 In this matrix product, the last m - r columns of E-l may be ignored together with the last m - r rows of U since these are identically zero. This produces a rn Comment. A factorization of a matrix A of rank r and size m x n corresponds to a factorization The preceding factorization may also be deduced as follows. Let us take any basis of the row space of A, say tl, . . . , t , . All rows may be expressed as linear combinations of these factorization A = EF'U, of the desired form. S A : R Z R, -+ R,. pi == s i l t 1 + ' ' ' + S i r t r = z l < k < , S i k t k .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Matrix Theory

  From Generalized Inverses to Jordan Form

  • Robert Piziak, P.L. Odell, Zuhair Nashed, Earl Taft(Authors)
  • 2007(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  For some matrices, the rank is easy to ascertain. For example, r ( I n ) = n . If A = diag ( d 1 , d 2 , ··· , d n ) , then r ( A ) is the number of nonzero diagonal elements. For other matrices, especially for large matrices, the rank may not be so easily accessible. We have seen that, given a matrix A , we can associate three subspaces and two dimensions to A . We have the null space N ull ( A ), the column space C ol ( A ), the row space R o w ( A ), the dimension dim ( N ull ( A )), which is the nullity of A , and dim ( C ol ( A )) = dim ( R o w ( A )) = r ( A ) , the rank of A . But this is not the end of the story. That is because we can naturally associate other matrices to A . Namely, given A , we have the conjugate of A , A ; the transpose of A , A T ; and the conjugate transpose, A ∗ = ( A ) T . This opens up a number of subspaces that can be associated with A : N ull ( A ) C ol ( A ) R o w ( A ) N ull ( A ) C ol ( A ) R o w ( A ) N ull ( A T ) C ol ( A T ) R o w ( A T ) N ull ( A ∗ ) C ol ( A ∗ ) R o w ( A ∗ ) . Fortunately, not all 12 of these subspaces are distinct. We have C ol ( A ) = R o w ( A T ), C ol ( A ) = R o w ( A ∗ ), C ol ( A T ) = R o w ( A ) , and C ol ( A ∗ ) = R o w ( A ) . Thus, there are actually eight subspaces to consider. If we wish, we can elim-inate row spaces from consideration altogether and just deal with null spaces and column spaces. An important fact we use many times is that rank ( A ) = rank ( A ∗ ) (see problem 15 of Exercise Set 8). This depends on the fact that if there is a dependency relation among vectors in C n , there is an equivalent dependency relation among the vectors obtained from these vectors by tak-ing complex conjugates of their entries. In fact, all you have to do is take the complex conjugates of the scalars that effected the original dependency relationship. We begin developing a heuristic picture of what is going on with the following diagram.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Mathematical Methods for Finance

  Tools for Asset and Risk Management

  • Sergio M. Focardi, Frank J. Fabozzi, Turan G. Bali(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  AP is a diagonal matrix where the diagonal is made up of the eigenvalues:

  SINGULAR VALUE DECOMPOSITION

  Suppose that the n ×m matrix A with m ≥ n has rank(A ) = r > 0. It can be demonstrated that there exists three matrices U , W , V such that the following decomposition, called singular value decomposition , holds:

  and such that U is n ×r with U ′ U = I r ; W is diagonal, with nonnegative diagonal elements; and V is m×r with V′V = I r .

  KEY POINTS

  • In representing and modeling economic and financial phenomena it is useful to consider ordered arrays of numbers as a single mathematical object.
  • Ordered arrays of numbers are called vectors and matrices; vectors are a particular type of matrix.
  • It is possible to consistently define operations on vectors and matrices including the multiplication of matrices by scalars, sum of matrices, product of matrices, and inversion of matrices.
  • Determinants are numbers associated with square matrices defined as the sum of signed products of elements chosen from different rows and columns.
  • A matrix can be inverted only if its determinant is not zero.
  • The eigenvectors of a square matrix are those vectors that do not change direction when multiplied by the matrix.
  • The column Rank of a Matrix is the maximum number of linearly independent column vectors of the matrix.
  • The row Rank of a Matrix is the maximum number of linearly independent row vectors of the matrix.
  • A matrix that has a rank as large as possible is said to have full rank; otherwise, the matrix is rank deficient.

  1. Variances and covariances are described in Chapter 6.

  2.

  Sign up to read

  Learn more about book