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What does it mean to multiply a vector by a matrix?
Let us define the multiplication between a matrix A and a vector x in which the number of columns in A equals the number of rows in x . So, if A is an m×n matrix, then the product Ax is defined for n×1 column vectors x . If we let Ax=b , then b is an m×1 column vector.
Why do we need matrix multiplication?
This are just simple rules to help you remember how to do the calculations. Rows come first, so first matrix provides row numbers. Columns come second, so second matrix provide column numbers. Matrix multiplication is really just a way of organizing vectors we want to find the dot product of.
What is the intuitive meaning of vector multiplication with covariance matrix?
Some intuition I already gathered is as follows: By multiplying Σ ∗ r we weight the samples X i according to r. With fixed i, y i gives us then the sum over the weighted covariances with X i and X j for 1 ≤ j ≤ n, which means a value of how well X i “covaries” in the direction of r.
How to calculate the entries of the covariance matrix?
With the covariance we can calculate entries of the covariance matrix, which is a square matrix given by C i, j = σ ( x i, x j) where C ∈ R d × d and d describes the dimension or number of random variables of the data (e.g. the number of features like height, width, weight, …).
What are some common intuitions for matrix multiplication?
What does matrix multiplication mean? Here’s a few common intuitions: 1) Matrix multiplication scales/rotates/skews a geometric plane. This is useful when first learning about vectors: vectors go in, new ones come out. Unfortunately, this can lead to an over-reliance on geometric visualization.
Which is the largest eigenvector of covariance matrix?
In a very brief statement, principal components in PCA are the eigen vectors of the covariance matrix which have large Eigen Values. Now consider the covariance matrix as the transformation matrix. Eigen vectors are the vectors whose direction will remain unchanged after multiplying it with the transformation matrix.