Both congruent and similarity of matrix are equivalence relations. The congruent and similar are different but relative. They cross at symmetric matrix

two symmetric matrices are congruent if and only if they are similar. Furthermore a quadric form can be changed to its standard form by orthogonal transformation. All tese are important contents in the advanced linear algebra. However the proof of this theory in existences is related to at least quadric form

linear space

linear transformation and Euclidean space. For the purpose of teaching

we give a new and explicit proof of this theory by using the orthogonal property of Euclidean space.