gibson:teaching:fall-2016:math753:norms-orthogonality
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gibson:teaching:fall-2016:math753:norms-orthogonality [2016/10/06 11:52] gibson |
gibson:teaching:fall-2016:math753:norms-orthogonality [2016/10/06 11:58] (current) gibson |
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| ====== Math 753/853 Norms, inner products, and orthogonality ====== | ====== Math 753/853 Norms, inner products, and orthogonality ====== | ||
| - | Ok, this is a big set of topics, and nothing I've found covers the topic at the right level of detail or depth. So, here's a summary of a few key points you should understand. | + | Ok, this is a big set of topics, and nothing I've found covers the topic at the right level of detail or depth. So, here's a summary of a few key points you should understand. These were spelled out in detail during lecture. |
| - | ===The inner product=== | + | ===Inner product=== |
| The inner product of two vectors $x$ and $y$ | The inner product of two vectors $x$ and $y$ | ||
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| where $x^T$ is the transpose of $x$. If $x^Ty = 0$, $x$ and $y$ are orthogonal. | where $x^T$ is the transpose of $x$. If $x^Ty = 0$, $x$ and $y$ are orthogonal. | ||
| - | + | ===2-norm=== | |
| - | ===The 2-norm=== | + | |
| The 2-norm of a vector $x$ is defined as | The 2-norm of a vector $x$ is defined as | ||
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| \|x\| = \sqrt{\sum_{i=1}^m x_i^2} | \|x\| = \sqrt{\sum_{i=1}^m x_i^2} | ||
| \end{equation*} | \end{equation*} | ||
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| + | Note that $x^Tx = \|x\|^2$. | ||
| The 2-norm of a matrix $A$ is defined as | The 2-norm of a matrix $A$ is defined as | ||
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| You can think of $\|A\|$ as the maximum amplification factor in length that can occur under the map $x \rightarrow Ax$. | You can think of $\|A\|$ as the maximum amplification factor in length that can occur under the map $x \rightarrow Ax$. | ||
| - | ===Orthogonal matrix=== | + | ===Orthogonal matrices=== |
| A matrix $Q$ is an orthogonal matrix if its inverse is its transpose: $Q^T Q = I$. The columns of an orthogonal matrix are a set of orthogonal vectors. | A matrix $Q$ is an orthogonal matrix if its inverse is its transpose: $Q^T Q = I$. The columns of an orthogonal matrix are a set of orthogonal vectors. | ||
| - | * | + | Key properties of orthogonal matrices: |
| + | * The inner product is preserved under orthogonal transformations: $(Qx)^T(Qy) = x^Ty$. | ||
| + | * The vector 2-norm is preserved under orthogonal transformations: $\|Qx\| = \|x\|$. | ||
| + | * The matrix 2-norm is preserved under orthogonal transformations: $\|QA\| = \|A\|$. | ||
| + | * The 2-norm of an orthogonal matrix is one: $\|Q\| = 1$. | ||
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| * [[https://en.wikipedia.org/wiki/Transpose | Transpose ]] | * [[https://en.wikipedia.org/wiki/Transpose | Transpose ]] | ||
| * [[https://en.wikipedia.org/wiki/Orthogonal_matrix | Orthogonal matrices ]] | * [[https://en.wikipedia.org/wiki/Orthogonal_matrix | Orthogonal matrices ]] | ||
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gibson/teaching/fall-2016/math753/norms-orthogonality.1475779973.txt.gz · Last modified: 2016/10/06 11:52 by gibson
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