A square matrix E∈Mn×n(R) is idempotent if E2=E. It is symmetric if E = tE.
(a) Let V⊆Rn be a subspace of Rn, and consider the orthogonal projection projV:Rn→Rn onto V. Show that the representing matrix E = [projV]EE of proj V relative to the standard basis E of Rn is both idempotent and symmetric.
(b) Let E∈Mn×n(R) be a matrix that is both idempotent and symmetric. Show that there is a subspace V⊆Rn such that E= [projV]EE. [Hint: What does the hypothesis that E is symmetric say about the relation between kerE and imE ?]
A square matrix E∈Mn×n(R) is idempotent if E2=E. It is symmetric if E = tE. (a) Let V⊆Rn be a subspace of Rn, and consider the orthogonal projection projV:Rn→Rn onto V. Show that the representing matr...
Please solve both parts of this question! I've stared at it for a long time without knowing how to approach it. (1) A square matrix E є м,xn(R) is idempotent if E-E. It is symmetric if -t E. (a) Let V C Rn be a subspace of R, and consider the orthogonal projection projy R" Rn onto V. Show that the representing matrix E = projy18 of proj v relative to the standard basis of IRn is both idempotent and...
(I) A square matrix E E M,xn(R) is idempotent if E-E. It is symmetric if E-E RR -[projyl& of projy relative to the standard basis (a) Let V C R be a subspace of R", and consider thé orthogonal projection projy onto V. Show that the representing matrix E & of IRn is both idempotent and symmetric. (b) Let E E Mnxn(R) be a matrix that is both idempotent and symmetric. Show that there is a subspace VCR" such that...
Find the orthogonal projection of v=[1 8 9] onto the subspace V of R^3 spanned by [4 2 1] and [6 1 2] (1 point) Find the orthogonal projection of v= onto the subspace V of R3 spanned by 2 6 and 1 2 9 projv(v)
Please show all work in READ-ABLE way. Thank you so much in advance. Problem 2.2 n and let X ε Rnxp be a full-rank matrix, and Assume p Note that H is a square n × n matrix. This problem is devoted to understanding the properties H Any matrix that satisfies conditions in (a) is an orthogonal projection matriz. In this problem, we will verify this directly for the H given in (1). Let V - Im(X). (b) Show that...
Problem 2. Recall that for any subspace V of R", the orthogonal projection onto V is the map projy : RM → Rn given by projy() = il for all i ER", where Ill is the unique element in V such that i-le Vt. For any vector space W, a linear transformation T: W W is called a projection if ToT=T. In each of (a) - (d) below, determine whether the given statement regarding projections is true or false, and...
(3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A- (3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A-
Let Y = (Yİ Y2 Yn)' be a random vector taking on values in Rn with mean μ E Rn and covariance matrix 2. Also let 1 be the ones vector defined by 1-(1 1) 5.i Find the projection matrix Hy where V is the subspace generated by 1 5.ii Show that Hy is symmetric and idempotent. 5.iii Let x = (a a . .. a)', where a E Rn. Show that Hvx = x. 5.iv Find the projection of...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
e, none of these 7. Let {1,..., up} be an orthogonal basis for a subspace W of R" and {...., } be an orthogonal basis for Wt. Determine which of the following is false. a. p+q=n b. {U1,..., Up, V1,...,0} is an orthogonal basis for R". c. the orthogonal projection of the u; onto W is 0. d. the orthogonal projection of the vi onto W is 0. e. none of these 8. Let {u},..., up} be an orthogonal basis...
2. Let A € Mn(R). (a) Show that AAT is a semipositive definite symmetric matrix and that AAT and AT A are similar. (b) Show by example that it need not be the case that AAT and ATA are similar for A E Mn(C).