charateristic polynomial of above matrix is
charateristic equation is
As 9 is a root so by long division method we have
Now we find eigenvector
we need to find orthogonal matrix Q that diagonalizes A
DEFINITION :- A n x n matrix
is orthogonal matrix if
where
is tranpose of
. In particular
i.e a matrix is invertible.
we have to find a orthogonal matrix
that diagonalizes A. In other words we are going to
write
where
is orthogonal matrix and
is diagonal.
therefore we normalise
now our
Linear algebra 4 5 5 (12 points) Consider the symmetric matrix A = 5 4 -5...
[12] QUESTION 4 (a) Let A be an m × m symmetric matrix and P be an orthogonal matrix such the PAP-D,where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that PA P is also a diagonal matrix. (b) Let A be an m × n matrix of rank m such that A = BC where B and C each has rank m. Show that (BC) CB. 16 STA4801/101/0/2019 (c) For the matrix...
[12] QUESTION 4 (a) Let A be an m × m symmetric matrix and P be an orthogonal matrix such the PAP-D,where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that PA P is also a diagonal matrix. (b) Let A be an m × n matrix of rank m such that A = BC where B and C each has rank m. Show that (BC) CB. 16 STA4801/101/0/2019 (c) For the matrix...
linear algebra
Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal -1 0-1 0-1 0 - 1 0 9 1 Find the characteristic polynomial of A. |x - Al- Find the eigenvalues of A. (Enter your answers from smallest to largest.) (21, 22, 23) Find the general form for every eigenvector corresponding to 21. (Uses as your parameter.) X1 - Find the general form for every elge vector corresponding to Az. (Uset as your...
The eigenvalues of the symmetric matrix A= ſi 8 41 8 1 -4 are 11 = 9 and 12 = -9. 14 -4 7 | Find an orthogonal diagonalization of A. Find the characteristic polynomial of A.
= ) 1 5 # 4 [10 points] Recall the real symmetric 2 by 2 matrix B of # 3. 5 1 # 4 a) From the matrix fact sheet, what are the eigenvalues of B and its characteristic polynomial ? # 4 b) Compute the trace of B, the sum of diagonal terms of the matrix B in three ways. 1- directly, 2- via eigenvalues, 3-via characteristic polynomial # 4 c) Compute the determinant of B, ad — bc,...
The symmetric matrix A below has distinct eigenvalues 10,-2 and-8. Find an orthogonal matrix P and a diagonal matrix D such that pTAP-Duse the square root symbol 'where needed to give an exact value for your answer. -1 47 A- 4 2-4 0 0 0] P=10 0 0| D=10 0 0
DETAILS LARLINALG8 7.3.033. Show that any two eigenvectors of the symmetric matrix A corresponding to distinct eigenvalues are orthogonal. 3 A = Find the characteristic polynomial of A. |u-A=1 Find the eigenvalues of A. (Enter your answers from smallest to largest.) (14, 12) = Find the general form for every eigenvector corresponding to 1. (Use s as your parameter.) X1 = Find the general form for every eigenvector corresponding to 12. (Use t as your parameter.) X2 = Find x,...
1. The symmetric matrix [4 1 1 1 A = 4-1 1 -1 4 -1 4 -1 has eigenvalues A = 1 (with algebraic multiplicity 1) and A 5 (with algebraic multiplicity 3). a) Find bases for the eigenspaces E(1) and E(5). b) Apply the Gram-Schmidt process to your basis for E(5) to find an onal basis for E (5) orthog- (c) Hence write down an that QT AQ = D. orthogonal matrix Q and a diagonal matrix D such...
1. For each of the following symmetric matrices, find an orthogonal matrix P and diagonal matrix D such that PTAP = D. 0 1 (а) А — 1 0 1 -1 1 0 2 -2 (Ъ) А %— -2 -2 -4 -2 2 |3 0 7 0 5 0 7 0 3 (с) А %—
1. For each of the following symmetric matrices, find an orthogonal matrix P and diagonal matrix D such that PTAP = D. 0 1 (а)...
Need help with linear algebra problem!
Let S be a symmetric, 2 x 2 matrix. Let û1) and ût2) be orthogonal eigenvectors of S with corresponding nonzero eigenvalues A1 and X2. Show that if v E R2 is a vector such that û1)Su = 0, then 5 = Bû(2) for some B 0.
Let S be a symmetric, 2 x 2 matrix. Let û1) and ût2) be orthogonal eigenvectors of S with corresponding nonzero eigenvalues A1 and X2. Show that...