Examples of identity matrix in the following topics:
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- The identity matrix $[I]$ is defined so that $[A][I]=[I][A]=[A]$, i.e. it is the matrix version of multiplying a number by one.
- The matrix that has this property is referred to as the identity matrix.
- The identity matrix, designated as $[I]$, is defined by the property:
- So $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is not an identity matrix.
- For a $3 \times 3$ matrix, the identity matrix is a $3 \times 3$ matrix with diagonal $1$s and the rest equal to $0$:
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- The matrix $B$ is the inverse of the matrix $A$ if when multiplied together, $A\cdot B$ or $B\cdot A$ gives the identity matrix.
- Note that, just as in the definition of the identity matrix, this definition requires commutativity—the multiplication must work the same in either order.
- The definition of an inverse matrix is based on the identity matrix $[I]$, and it has already been established that only square matrices have an associated identity matrix.
- When multiplying this mystery matrix by our original matrix, the result is $[I]$.
- This is called a singular matrix.
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- The summation in the Fourier interpolation is therefore a matrix-vector inner product.
- Let's identify the coefficients of the matrix.
- Define a matrix $Q$ such that
- The matrix $Q$ is almost orthogonal.
- We have said that a matrix $A$ is orthogonal if $A A^T = A^T A = I$, where $I$ is the N-dimensional identity matrix.
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- Look under the Tools>Matrix Algebra menu.
- If you do know some matrix algebra, you will find that this tool lets you do almost anything to matrix data that you may desire.
- This is a mathematical operation that finds a matrix which, when multiplied by the original matrix, yields a new matrix with ones in the main diagonal and zeros elsewhere (which is called an identity matrix).
- Now suppose that we multiply this adjacency matrix times itself (i.e. raise the matrix to the 2nd power, or square it).
- This matrix (i.e. the adjacency matrix squared) counts the number of pathways between two nodes that are of length two.
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- One is, what to do with the items in the similarity matrix that index the similarity of an actor to themselves (i.e. the diagonal values)?
- If the data being examined are symmetric (i.e. a simple graph, not a directed one), then the transpose is identical to the matrix, and shouldn't be included.
- If you are working with a raw adjacency matrix, similarity can be computed on the tie profile (probably using a match or Jaccard approach).
- The first panel shows the structural equivalence matrix - or the degree of similarity among pairs of actors (in this case, dis-similarity, since we chose to analyze Euclidean distances).
- That is, there are no two cases that have identical ties to all other cases.
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- The original data matrix has been reproduced below as figure 13.2.
- Many of the features that were apparent in the diagram are also easy to grasp in the matrix.
- This means that the entries in the rows and columns for one actor are identical to those of another.
- If the matrix were symmetric, we would need only to scan pairs of rows (or columns).
- The goal here is to create an actor-by-actor matrix of the similarity (or distance) measures.
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- The identity element in the space of square $n \times n$ matrices is a matrix with ones on the main diagonal and zeros everywhere else
- As an exercise, show that $A I_n = I_n A = A$ for any $n\times x$ matrix $A$ .
- We can take any vector in $\mathbf{R}^{n}$ and make a diagonal matrix out of it just by putting it on the main diagonal and filling in the rest of the elements of the matrix with zeros.
- An orthogonal matrix has an especially nice geometrical interpretation.
- Therefore an orthogonal matrix maps a vector into another vector of the same norm.
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- are dependent — they are the same equation when scaled by a factor of two, and they would produce identical graphs.
- $egin{matrix} x-2y &= &-1\ 3x+5y &= &8\ 4x+3y &=& 7 nd{matrix} $
- $egin{matrix} x+y &= &1\ 2x+y &= &1\ 3x+2y &=& 3 nd{matrix} $
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- The first step in examining structural equivalence is to produce a "similarity" or a "distance" matrix for all pairs of actors.
- This matrix summarizes the overall similarity (or dissimilarity) of each pair of actors in terms of their ties to alters.
- A number of methods may be used to identify patterns in the similarity or distance matrix, and to describe those patterns.
- Groupings of structurally equivalent actors can also be identified by the divisive method of iterating the correlation matrix of actors (CONCOR), and by the direct method of permutation and search for perfect zero and one blocks in the adjacency matrix (Optimization by Tabu search).
- The structural equivalence concept aims to operationalize the notion that actors may have identical or nearly identical positions in a network -- and hence be directly "substitutable" for one another.
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- If the adjacency matrix for a network can be blocked into perfect sets of structurally equivalent actors, all blocks will be filled with zeros or with ones.
- If two actors have identical geodesic distances to all other actors, they are (probably) automorphically equivalent.
- Why does having identical distances to all other actors make actors "substitutable" but not necessarily structurally equivalent?
- Make an adjacency matrix for a simple bureaucracy like this.
- Block the matrix according to the regular equivalence sets; block the matrix according to structural equivalence sets.