Physical significance of orthogonal matroces
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Firstly, please bear in mind that I do not have much knowledge about linear algebra. Secondly, I am not asking about some mathematical definitions but rather a more physical definition of this mathematical meaning.
So, the problem is that I can understand the meaning of orthogonality between two vectors, they are just "lines" perpendicular to each other, but I can not physically perceive what orthogonality means for matrices.(like a 3×3 one). I mean if vectors are like lines in space,those being orthogonal is an easy concept to visualize.But how to visualize two orthogonal matrices? I think that this is easier to explain to me if you first explain how does a matrix look like in space(visualization).(for example,for a 2x2,is it the area that its two contained vectors form in space?If it is that,then two orthogonal matrices are simply those two areas that are perpendicular to each other.If not,give me an analogous explanation).
So, the problem is that I can understand the meaning of orthogonality between two vectors, they are just "lines" perpendicular to each other, but I can not physically perceive what orthogonality means for matrices.(like a 3×3 one). I mean if vectors are like lines in space,those being orthogonal is an easy concept to visualize.But how to visualize two orthogonal matrices? I think that this is easier to explain to me if you first explain how does a matrix look like in space(visualization).(for example,for a 2x2,is it the area that its two contained vectors form in space?If it is that,then two orthogonal matrices are simply those two areas that are perpendicular to each other.If not,give me an analogous explanation).
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