Let A be an n × n matrix and let B = A − αI for some scalar α. How do the eigenvalues of A and B compare? Explain.

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 14 and 5. 9 4 2 4 9 2 2 2 6 Enter the matrices P and D below. (Use a comma to separate matrices as needed. Type exact answers, using radicals as needed. Do not label the matrices.)

Please show all work clearly, thank you.

Let A be a 3 x 3 diagonalizable matrix whose eigenvalues are X₁ = 3, A₂ = −2, and A3 = -1 with corresponding eigenvectors @· --D 0 1 0 Express A as PDP-1 where D is a diagonal matrix and use this to find A5. V₁ = A5 = V3 = 1, respectively.

Please try to explain by working backwards from given answer.

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are - 10, 2, and - 2. - 4 - 4 - 2 A = - 4 -2 - 4 2 -4 - 4 Enter the matrices P and D below. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)

For matrix A, find the associated eigenvalues. A = [] O2 and 3 O3 and 5 O1 and 5 5 and -1

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Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 4 and - 14. - 4 - 4 8 A = - 4 2 4 4 - 4 Enter the matrices P and D below. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)

Find all the eigenvalues (real and complex) of the matrix The eigenvalues are M = -9 12 3 -3 6 -2 -12 12 6 (Enter your answers as a comma separated list.)