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Let $B$ be an $n \times n$ symmetric matrix such that $B^{2}=B .$ Any such matrix is called a projection matrix (or an orthogonal projection matrix). Given any $y$ in $\mathbb{R}^{n},$ let $\hat{\mathbf{y}}=B \mathbf{y}$ and $\mathbf{z}=\mathbf{y}-\hat{\mathbf{y}}$ a. Show that $\mathbf{z}$ is orthogonal to $\hat{\mathbf{y}}$b. Let $W$ be the column space of $B$ . Show that $y$ is the sum of a vector in $W$ and a vector in $W^{\perp} .$ Why does this prove that $B y$ is the orthogonal projection of $y$ onto the columnspace of $B ?$

Any vector in $\mathrm{W}=\mathrm{Col} \mathrm{B}$ has the form Bu for some u. Noting that $\mathrm{B}$ is symmetric, Exercise 28 gives $(y \mathrm{y} \text { (hat) }) \cdot(\mathrm{Bu})=[\mathrm{B}(\text { yy (hat) })] \cdot \mathrm{u}=[\text { ByBBy }] \cdot \mathrm{u}$ $=0$ since $\mathrm{B} 2=\mathrm{B}$ . So yy (hat) is in $\mathrm{W}$ perp, and the decomposition $\mathrm{y}=$ y(hat) $+(\text { yy (hat) ) expresses } \mathrm{y} \text { as the sum of a vector in } \mathrm{W} \text { and a vector }$ in W perp. By the Orthogonal Decomposition Theorem in Section 6.3 , this decomposition is unique, and so y (hat) must be the projection of y onto W.

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 1

Diagonalization of Symmetric Matrices

Introduction to Matrices

Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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you're given a special kind of matrix and were asked to prove statements about this matrix were given that B is an n by n symmetric matrix such that B squared is equal to be. And we're told that this is a projection matrix what this is called or an orthogonal projection matrix in some cases. And we're told that if why is some victor and are in will denote my white hat matrix speed times the vector Why So why had is another vector and Z will be why minus y hat which is another vector In part they were asked to show that z is orthogonal toe y hat Do this we have z started with why had is equal to well, we can write move these in terms of why so Z is equal to why minus y hat started with and then why hat is equal to B? Why by definition and this is then equal to why minus b why started with you? Why again and we have by the ad activity of the dot product of this is equal to why started with b y minus b y started with do you? Why so we have. This is equal to why started with B, Why minus and then since B is an n by n symmetric matrix and that this is going to be equal to I had previous exercise Why dotted with B oth be of why or be times B times by This is equal to why that it would be why minus why dotted with he's squared. Why recall that B squared is equal to be so. This is equal to why started with b y minus why dotted with B y Elizabeth scaler so simply reduces to zero. And so we've shown that Z and white hat are orthogonal in part b were given that w is the column space of B and were asked to show that why is the sum of a vector in W and a vector in W perp or because this w compliment. So prove this. We have that why is some vector and are in And so why? Why Hat started with BU. So we have that bu is gonna be an element w for some Dr you in Oran. It's what means to say that it's in the column space and this is equal to again since music symmetric matrix This is the same as B Why? Why Hat started with you which is equal to recall that why hat is B y says his be times Why times e times why died with you And we can regret this Since b squared is equal to be as be Why be be why started with you we have this is equal to transpose of e b why you is equal to by the laws of transposition Why transpose the transposed be transposed Why transposed be transposed times you and we have that since he is a symmetric matrix This is equal to why transpose b times be by transposed being times You gonna revise a little bit My proof here so actually left out something important this should be instead of why Why had this is why minus y hat bu which can then be written as b times y minus y hat Just say mus be wind minus the white hat dot you which is unequal to B Y minus. This becomes b squared. Why definition of white hat tied with you and this is then going to be equal to by the activity of the dot product. But first, this is equal to be why minus B Why, since B squared equals B started with you just simply equal to zero dotted with you is your vector, That is which is zero. So you've shown that you is orthogonal to y minus Why hat? Which means that why minus y hat since its orthogonal to every vector in the calls base of B is a member of compliment of the column space of B Therefore, we have that blind minus Why had is equal to some Decter. I'll call it the where V is in the complement of W so that why is equal to why hat plus the where we have that why hat is in the calm space of be and V hats in the complement of the columns Basit B. And we have that in particular the is just equal to the vector. Why minus why hat Next? We're asked, why does this prove that B y is the orthogonal projection of why onto the column space of B. Well, we have that by the earth ogle decomposition here. Um, this is a unique decomposition and so it follows that. Why hat speak well to projection of why on to w So we have that. B y is the projection of why onto Collins base of B, which is W.

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