Matrix Limit

• Let . The sequence is said to converge to the matrix called the limit of the sequence if

  We denote this by saying that

• (Frieidberg 5.17) Let such that

  Then for any and we have that

• (Friedberg 5.17.1) Let and . Then for any invertible matrix we have

• Let . This set consists of the complex number and the interior of the disk of radius in the complex plane.

  • If is a complex number, exists if and only if

  • (Friedberg 5.18) Let be a square matrix with complex entries. Then exists if and only if the following hold

    • If is an eigenvalue of , then
    • If is an eigenvalue of then the dimension of the eigenspace corresponding to equals the algebraic multiplicity of as an eigenvalue of .

  • (Friedberg 5.19) Let be such that the following hold If is an eigenvalue of , then is diagonalizable.

    Then exists.

    • This gives us a technique for computing matrix limits on diagonalizable matrices — first express the matrix as then compute the limit as shown in (Friedberg 5.17)

• Let . is the sum of the absolute values of the entries of row . is the sum of the absolute values of the entries of column

  The row sum and the column sum are defined as

• (Friedberg 5.21, Mesbahi 3.9) Gerschgorin’s Disk Theorem. Let define

  and let denote the disk centered at of radius . Then each eigenvalue of lies in some . More formally, each eigenvalue is located in

  • (Friedberg 5.21.1) Let be any eigenvalue of . Then
  • (Friedberg 5.21.2) Let be any eigenvalue of . Then

• (Friedberg 5.23) Let be a matrix in which each entry is positive and let be an eigenvalue of such that . Then and is a basis for where

  • (Friedberg 5.23.1) Let be a matrix in which each entry is positive and let be an eigenvalue of such that . Then and

• If , we define as

  Thus, is the sum of the infinite series

  • (Friedberg e5.3.19) Let be a diagonal matrix. Then

Links §

• Friedberg, Insel and Spence

• Mesbahi and Egerstedt

• Real Analysis

• Matrix