Question

Exercise 5 a) Let A be a diagonalizable $n \times n$ matrix that has only $\lambda = 1$ and $\lambda = -1$ as eigenvalues. Show that $A^2$ is $I_n$, where $I_n$ is the $n \times n$ identity matrix. b) Let A be an $m \times n$ matrix, and let $K = A^tA$. Show that the identity $Nul(A) = Nul(K)$ holds for any choice of A.

          Exercise 5
a) Let A be a diagonalizable $n \times n$ matrix that has only $\lambda = 1$ and $\lambda = -1$ as eigenvalues. Show that $A^2$ is $I_n$, where $I_n$ is the $n \times n$ identity matrix.
b) Let A be an $m \times n$ matrix, and let $K = A^tA$. Show that the identity $Nul(A) = Nul(K)$ holds for any choice of A.

Exercise 5
a) Let A be a diagonalizable n × n matrix that has only λ = 1 and λ = -1 as eigenvalues. Show that A^2 is In, where In is the n × n identity matrix.
b) Let A be an m × n matrix, and let K = A^tA. Show that the identity Nul(A) = Nul(K) holds for any choice of A.

Added by Jennifer G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shima Shaw

04/22/2022

Step 1

Step 1: Since A is diagonalizable, it can be written as A = PDP^-1, where D is a diagonal matrix with the eigenvalues of A on the diagonal and P is the matrix whose columns are the corresponding eigenvectors. Show more…

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Exercise 5 a) Let A be a diagonalizable n x n matrix that has only X = 1 and X = --1 as eigenvalues. Show that A2 is In, where In is the n n identity matrix. b) Let A be an m x n matrix, and let K = AtA. Show that the identity Nul(A) = Nul(K) holds for any choice of A.