Question

find diagonalization of matrix of first row[ 1 1 3] & second row [1 5 1] & third row is [3 1 1]? ​

Answer 1

Answer:

$Let A be an n� n matrix over a field F . We recall that a scalar l Î F is said to be an eigenvalue (characteristic value, or a latent root) of A, if there exists a nonzero vector x such that Ax = l x, and that such an x is called an eigen-vector (characteristic vector, or a latent vector) of A corresponding to the eigenvalue l and that the pair (l , x) is called an eigen-pair of A. If l is an eigenvalue of A, the equation: (l I-A)x = 0, has a non-trivial (non-zero) solution and conversely. Thus, this being a homogeneous equation, it follows that l is an eigenvalue of A iff |l I-A| = 0. The expression</p><p></p><p>fA(x) = |xI-A|</p><p></p><p>is a monic (the coefficient of the highest power of x in it is 1) polynomial in x of degree n. It is known as the characteristic polynomial of A. Thus l is an eigen value of A iff l is a zero (or root) of the characteristic polynomial fA(x) in F. The equation</p><p></p><p>fA(x) = 0,</p><p></p><p>$