Hi Nugatory,Nugatory said:In a two-dimensional space such as the surface of a sheet of paper you can only have two mutually perpendicular axes. In a three-dimensional space you can have three. You cannot visualize a four-dimensional space, but mathematically it's a perfectly sensible concept - and how many axes do you need there?
You just need a definition of "perpendicular" that's not tied to your ability to visualize 90-degree angles.
Consider the three dimensional case. Pick any two axis and ignore the third. Those two are in a 2 dimensional plane and the axes are at right angles.ss k said:Hi Nugatory,
Thanks usually there are n-vectors so n-axes and all are perpendicular with respect to each other. I don't want to plot them or see in some 3-d graphics programme but just wondering how they can even exist. Let's say I have 4 such axes (having values from -,0,+ and center orgin at 0) then axis 1,2,3 can be perpendicular with each other and how come the 4th axis will be perpendicular to any other axis.
Perpendicular eigenvectors are vectors that are at right angles to each other. This means that when the two vectors are plotted, they will form a 90 degree angle. In linear algebra, perpendicular eigenvectors are important because they are independent of each other and can be used to transform a matrix into its diagonal form.
Having more than 3 perpendicular eigenvectors is important because it allows for a more precise transformation of a matrix into its diagonal form. The more eigenvectors that are perpendicular, the more independent directions there are to transform the matrix in. This can be useful in many applications, such as image processing and data compression.
Yes, a matrix can have more than 3 perpendicular eigenvectors. In fact, there is no limit to the number of perpendicular eigenvectors that a matrix can have. However, the number of perpendicular eigenvectors is limited by the size of the matrix. For example, a 3x3 matrix can have a maximum of 3 perpendicular eigenvectors.
To find perpendicular eigenvectors, you can use the eigenvalue-eigenvector equation: Av = λv. First, find the eigenvalues of the matrix. Then, for each eigenvalue, find the corresponding eigenvector. Finally, check if the eigenvectors are perpendicular to each other by calculating their dot product. If the dot product is 0, then the eigenvectors are perpendicular.
Eigenvectors that are perpendicular to each other are essential in diagonalizing a matrix. This is because the eigenvectors form a basis for the matrix, and when they are perpendicular, they can be used to transform the matrix into its diagonal form. Diagonalizing a matrix is important in many applications, such as solving systems of linear equations and finding the eigenvalues of a matrix.