Gram–SchmidtRecipe: (Gram–Schmidt orthonormalization)Given a basisa1,, an, produce an orthonormal basis q1,, qn.b1= a1, q1=b1kb1kb2= a2− ha2, q1iq1, q2=b2kb2kExample 2. Find an orthonormal basis for V = span1000,2100,1111.Solution.Armin [email protected] 3. An orthogona l matrix is a square m atrix with orthon ormal columns.Theorem 4. An n × n matrix Q is orthogonalQTQ = IProof. Example 5. Q =cos θ −sin θsin θ cos θQis orthogonal becauseThe QR dec ompositio n (fla shed at you)Let A be an m × n matrix of ra nk n. (columns independent)Then we have the QR decompositionA = QR,• where Q has orthonormal columns, and• R is upper triangular and invertible.Idea: Gram–Schmidt on the columns o fA, to ge t the columns of Q!Example 6. Find the QR decomposition of A =1 2 40 0 50 3 6.Armin [email protected] general, A = QR is obtained as:| |a1a2| |=| |q1q2| |ha1, q1i ha2, q1i ha3, q1iha2, q2i ha3, q2iha3, q3iPractice problemsExample 7. Find the QR decomposition of A =1 1 20 0 11 0 0.Armin
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