Induced Norm-Matrix p-Norms Induced norm-Matrix p-norms Let AeCmxn. The p-norm IAllp max Axlp *0 xp p21, is a matrix norm. Equivalently,the p-norm of a matrix can be rewritten as IAp=max lAxllp,p≥1， x p=1 Remarks:The matrix p-norms are based on the vector p-norms and measure how much a matrix can stretch a unit-norm vector. 务这头子 Matrix Theory Matrix Norms -9/35
Induced Norm–Matrix p-Norms Induced norm–Matrix p-norms Let A ∈ C m×n . The p-norm ∥A∥p = max x≠0 ∥Ax∥p ∥x∥p , p ≥ 1, is a matrix norm. Equivalently, the p-norm of a matrix can be rewritten as ∥A∥p = max ∥x∥p=1 ∥Ax∥p, p ≥ 1, Remarks: The matrix p-norms are based on the vector p-norms and measure how much a matrix can stretch a unit-norm vector. Matrix Theory Matrix Norms - 9/35
Induced Norm-Matrix p-Norms Remark The matrix p-norms are extremely useful because they satisfy the submultiplicative inequality. 奇电有这头 Matrix Theory Matrix Norms -10/35
Induced Norm–Matrix p-Norms Remark The matrix p-norms are extremely useful because they satisfy the submultiplicative inequality. Matrix Theory Matrix Norms - 10/35
Induced Norm-Matrix p-Norms Remark The matrix p-norms are extremely useful because they satisfy the submultiplicative inequality. Let Ae Cmxn and yeC.Then IAylp≤Alplylp- 命电有这女 Matrix Theory Matrix Norms -10/35
Induced Norm–Matrix p-Norms Remark The matrix p-norms are extremely useful because they satisfy the submultiplicative inequality. Let A ∈ C m×n and y ∈ C n . Then ∥Ay∥p ≤ ∥A∥p∥y∥p. Matrix Theory Matrix Norms - 10/35
Induced Norm-Matrix p-Norms Remark The matrix p-norms are extremely useful because they satisfy the submultiplicative inequality. Let Ae Cmxn and yC.Then Ayle≤Alplylp. Tips: Clearly true, y=0 兜≥恍，y0 Allp maxx+0 xip 奇电有头 Matrix Theory Matrix Norms -10/35
Induced Norm–Matrix p-Norms Remark The matrix p-norms are extremely useful because they satisfy the submultiplicative inequality. Let A ∈ C m×n and y ∈ C n . Then ∥Ay∥p ≤ ∥A∥p∥y∥p. Tips: ⎧⎪⎪ ⎨ ⎪⎪⎩ Clearly true, y = 0; ∥A∥p = maxx≠0 ∥Ax∥p ∥x∥p ≥ ∥Ay∥p ∥y∥p , y ≠ 0. Matrix Theory Matrix Norms - 10/35
Induced Norm-Matrix p-Norms One Norm One norm-the maximal absolute column sum What if p=1? 奇电有这头 Matrix Theory Matrix Norms -11/35
Induced Norm–Matrix p-Norms One Norm One norm–the maximal absolute column sum What if p = 1? Matrix Theory Matrix Norms - 11/35