Eigenvalue Equality & Matrix Invertibility

Related BrainMass Solutions

• Linear Algebra : Invertible Matrices

  Thus we have which is in the form of , where is the eigenvalue of . The link between invertibility of matrices and eigenvalues is investigated. The solution is detailed and well presented.

• Linear Algebra : Zero Matrix Proof

  The invertibility of a matrix function is given.

• Invertibility of Matrices

  49929 Invertibility of Matrices Can you please explain to me why the columns of the nxn matrix A span R^n when A is invertible?

• Proof: 2x2 matrix invertible iff det not equal to 0

  Additionally, it provides proof that nonzero determinant implies invertibility. The solution provides two examples of these proofs.

• Minimax Principle - Intermediate Eigenvalue

  Use the minimax principle to show that the intermediate eigenvalue of is not positive. Hint: The matrix has a negative, a zero, and a positive eigenvalue.

• Idempotent and nilpotent matrix proofs

  If is a nilpotent matrix, then for some positive integer . Suppose is an eigenvalue and is the corresponding eigenvector, then we have . Then we have This implies that . So the only eigenvalue of a nilpotent matrix is 0.

• eigenvalue and eigenvector of matrix

  So A only changes the length of v and thus it proves that equation (*) holds true for eigenvalue To prove that λ is an eigenvalue of matrix, we then use the definition Therefore, λ = - 2 is an eigenvalue of matrix A.

• Solving for Eigenvalues

  So we have We get the characteristic polynomial of is Note: If A has an inverse matrix and has an non-zero eigenvalue a, then 1/a is an eigenvalue of its inverse matrix. You can prove it easily.

• A Simple introduction to eigenvectors and eigenvalues

  Wasn't -1 the eigenvalue, and (4,1) the eigenvector. So it worked :) Let's try one more. Using the same matrix, let's prove that -2 is an eigenvalue, and (3, 1) is an eigenvector. So here we go.