Multiply Matrix by Vector in Matlab

When working with matrices and vectors in Matlab, it's essential to understand how they can be multiplied together. However, there are some important considerations to keep in mind when doing so.

Firstly, it's crucial to remember that 1xN matrices are not equivalent to N-element vectors. This is because Matlab will attempt to solve a linear system for square matrices, whereas for rectangular matrices, it will seek the least squares solution.

Fortunately, Julia also has built-in operators for solving matrix operations. For example, you can perform dot products using the `dot` function or use the backslash operator (`\`) in a typical linear algebra notation.

In Matlab, the recommended approach is to multiply your matrix by the vector to obtain all result vectors at once. This takes advantage of optimized vector and matrix multiplication, which can significantly improve performance.

For element-wise multiplication, you can use the dot operator (`.`) to broadcast scalar multiplication across an array. Alternatively, you can use a Matlab-style syntax for matrix-matrix multiplication using the `*` operator.

Matrix and vector multiplication can also be performed using the `*` operator in Julia. Additionally, two-dimensional arrays (or matrices) are a fundamental part of Julia, and understanding how to operate on them is crucial for effective programming.

When multiplying vectors and matrices together, it's essential to consider the performance implications. For example, some computers may be able to execute multiple instructions at once, which can improve performance. However, others may have limitations, such as my own computer which can only perform 2 floating-point operations per cycle using single precision arithmetic.

Fortunately, Julia and Matlab both provide optimized implementations of matrix multiplication that can take advantage of these capabilities. In fact, I was able to achieve matrix multiplication speeds of over 30 GigaFlops on my CPU using Matlab's built-in operators, which is significantly faster than what would be possible using a simple loop-based implementation.

In conclusion, multiplying matrices and vectors in Matlab or Julia requires careful consideration of the underlying mathematics and the performance implications. By understanding how to use the built-in operators and syntax effectively, you can write efficient and effective code that takes advantage of optimized matrix multiplication algorithms.

Multiply matrix by vector matlab